If the spacetime holds for n\le 3 why not to present it as a nxn matrix with number of columns/rows n\ge 3 and calculate as a linear combination of multicomponent (multidimensional vectors)?

I address only the question "Can Space--times Exist in arbitrary Dimensions?" The short answer is yes, why not? The spacetime (which does not distinguish between time and space) underlying our world, characterised by time plus three-dimensional space, can be modeled with an algebra (spacetime algebra, STA) over a real 4D Minkowski vector space with signature (1,3). The even sub-algebra of this STA is associated with our world of a 3D space plus time. Time and space as we perceive them are obtained from the 4D algebra by means of a (purely algebraic) space-tme split. In a world of arbitrary spatial dimensions, n, the spacetime has signature (1,n) and (n+1) dimensions. For more detail, see the paper by David Hestenes: Spacetime Physics with Geometric Algebra, Am. J. Phys, 71 (6), June 2003.

Not physically.

In math space is anything you like.

Physics is a science and as such uses mathematical methods

Yes, space-time can exist in arbitrary dimensions according to theoretical physics. In models like string theory and M-theory, space-time is often described as having more than the familiar four dimensions (three spatial dimensions and one time dimension). These theories propose that additional dimensions—possibly up to 10 or 11—exist beyond our perception, though they may be compactified or hidden at scales beyond current experimental detection. The concept of space-time in arbitrary dimensions is a theoretical extension that allows for richer models of fundamental forces and particles.