The original Kaluza-Klein theory attempted exactly this, namely using the additional degrees of freedom in 5-dimensional spacetime to describe the electromagnetic field. There have been further attempts to generalize the theory to include other forms of matter (see "Space-Time-Matter theory", advocated by Paul Wesson and colleagues).

Another approach is to generalize the metric tensor to a nonsymmetric tensor, which can be split into a symmetric and an antisymmetric part. The symmetric part would represent gravity, while the antisymmetric part would be the Maxwell tensor of electrodynamics. This is the unified field theory approach that Einstein pursued late in his life. A good (albeit hard to find) monograph on this subject is Tonnelet's "Einstein's Theory of Unified Fields".

These classical attempts to unify gravity with electromagnetism and possibly other forces are not popular these days as there is no clear connection between them and quantum field theory, which in turn forms the basis of the standard model of particle physics, which, while somewhat inelegant (18+ free parameters), has been enormously successful at predicting and explaining all known forms of matter and interactions except gravity.

The original Kaluza-Klein theory attempted exactly this, namely using the additional degrees of freedom in 5-dimensional spacetime to describe the electromagnetic field. There have been further attempts to generalize the theory to include other forms of matter (see "Space-Time-Matter theory", advocated by Paul Wesson and colleagues).

Another approach is to generalize the metric tensor to a nonsymmetric tensor, which can be split into a symmetric and an antisymmetric part. The symmetric part would represent gravity, while the antisymmetric part would be the Maxwell tensor of electrodynamics. This is the unified field theory approach that Einstein pursued late in his life. A good (albeit hard to find) monograph on this subject is Tonnelet's "Einstein's Theory of Unified Fields".

These classical attempts to unify gravity with electromagnetism and possibly other forces are not popular these days as there is no clear connection between them and quantum field theory, which in turn forms the basis of the standard model of particle physics, which, while somewhat inelegant (18+ free parameters), has been enormously successful at predicting and explaining all known forms of matter and interactions except gravity.

The electromagnetic field may be interpreted as a curvature in gauge theories, where the potentials are the connections. The derivative of the dual of the electromagnetic field follows the Bianchi identity as it happens with the Riemann tensor. The symmetry associated to the global gauge is the continuous one dimensional group U(1), which can be generalized to the non-abelian SU(N) for generalizing the conserved ( electric charges, quarks with color, flavour, etc).

Geometrically even you can glue them with gravitation associated to the space-time symmetries employing fiber bundles. Say the geometrical interpretation is got nowadays, and what lacks is the unification. Kaluza-Klein used new concepts as unified compactification trying to unify them but it was not successful and the main problem is that Einstein´s gravitation is against this unification.

Sorry, spontaneous compactification, of couse.

Space does not undergo curvature. How gravity works, kindly see my article

http://www.eioba.com/a/4f0v/gravity-how-to-understand-it

Good question!  To my knowledge Weyl's paper "Gravitation and Electricity" from 1918 is the closest to treating electromagnetism in the way the GR treats the gravitational force.  Baldomir also mentions some later work building on this idea.

    The theory of unified gravitation and electromagnetism in a geometrical scope introduced by Weyl was really fantastic. In fact he pointed the importance of the gauge transformations of electrodynamics (not a mere mathematical trick). Unfortunately they were not related directly with the external spacetime symmetries, but with the inner U(1) symmetries that shares electrodynamics and quantum mechanics, i.e. the function associated to the gradient of the potentials corresponds to an angle of the quantum states in a geometrical picture associated to the U(1) differential form. It took almost 50 years to realize this fact, but unfortunately it was not enough for glue electrodynamics with gravitation in this way. 

Having just re-read the relevant sections in Weyl's book, "Space-Time-Matter", my understanding is that his proposal was to treat the metric (or rather, the "quadratic groundform", $g_{\mu\nu}dx^\mu dx^\nu$) and the 4-vector field of electromagnetism (as it appears in the "linear groundform" $\phi_\mu dx^\mu$) conceptually on an equal footing, both representing the geometry of spacetime. So electromagnetism is not something that happens "inside" spacetime, it is as much part of the geometry of spacetime as gravity is.

Weyl coined the term "gauge transformation" as to a change in the scale for the spacetime. In fact Weyl used the Riemannian metric multiplied by a function (electromagnetic potentials) which is associated to the change of scale (conformal transformations). And it is true that what we understand nowadays by gauge theories are applied in a very different form, i.e. to inner symmetries and no to SO(3,1) one.

In 1929 he recognized that his theory was wrong and published a paper in Zeitschrift fur Physik that tried to make a connection with quantum mechanics after the publication of London:

London, F. Quantum Mechanical Interpretation of Weyl's Theory

. Zeit. f. Phys. 42 (1927)

Therefore in spite of being a wrong theory of unification and also unable to be interpreted with quantum fields, his importance in theoretical physics was very important for making the geometry of the gauge theories.