I don't really think that representation theory is required here, it's easier than that. (Although representation theory is really cool!!)

Every finite dimensional vector space over the real numbers is isomorphic to R^n, for some n. But there is not always a canonical or "unique" isomorphism. I think the real difference -- and it is a subtle one -- is that R^n always comes with an "obvious" choice of basis, and many vector spaces don't.

I think the following may be the easiest interesting example. Consider the subspace of R^3, consisting of all (x, y, z) for which

x + y + z = 0.

This is a 2-dimensional real vector space. It is isomorphic to R^2, although in some sense it does not "present as such": you have to choose an isomorphism to R^2 if you want to "treat it as R^2", and there is no single choice that stands out.

In practice, one would not necessarily construct an isomorphism to R^2, or (more or less equivalently) exhibit a basis, before working with this vector space directly.

I think you could do some interesting stuff with error-correcting codes or polynomial spaces (or more generally function spaces).

I'm not too familiar with representation theory. Do you know where I could find a worked example of constructing, say, a representation of (F_p)^m on R^n? (for prime p and m>1)

Visually, (F_p)^m can be embedded in R^m trivially, but I suspect you are interested in animations of operations on F^p, which indeed would look different from R^m due to modular arithmetic, and so would be beyond the scope of the OP.