The stiffness matrix is same regardless of the solver. You could use matrix output command in standard to get stiffness matrix

The advantages of the explicit solver is that only the residual {r}  (right side) is computed and, in the most cases, the diagonalized (lumped) matrix is employed [M_d] for the solution of the dynamic equations  [M] \ddot {u} + [K] {u} = {f}. The most efficient explicit  code contains just {r} =  {f} - [K] {u}, computed at the previous time step in which the stiffness matrix is not computed at all. (this is the advantage of the explicite code, because you do not need the expensive linearization of the sophisticated material law. These rules are used in explicit Abaqus as well as in explicit LS-DYNA. That is why the stiffness matrix is not generated at the particular time step (and there is no any iteration). If you are using IMPLICIT then the tangent /stiffness/ matrix is the most important part and, of course, is computed at each time step as well as (depending on the solver) at each iteration (solution is iterative and may or may not converge).