I have taken several linear algebra courses, one from my high school and two from universities. The thing is, not all courses of linear algebra will discuss rotations the way you discuss it. One reason is that sometimes a high school linear algebra course cannot assume students have learned trigonometry. I've seen teachers teach it just to solve larger linear systems of equations. Another reason is that sometimes a course will focus just on properties of vector spaces without relating them to geometry; after all who can visualize things when the course routinely deals with 10-dimensional vectors or N-dimensional ones where N isn't a constant.
I think teaching beginner linear algebra using matrices representing systems of equations is a pedagogical mistake. It gives the wrong impression that matrices are linear algebra and makes it difficult for students to think about it in an abstract way. A better way is to start by discussing abstract linear combinations and then illustrating what can be done with this using visualizations in various coordinate systems. Once the student understands this intuitively, systems of equations and matrices can be brought up as equivalent ways to represent linear transformations on paper. It’s important to emphasize that matrices are convenient but not the only way to write the language of linear algebra.
Usually you just draw a 2D or 3D picture and say "n" while pointing to it. e.g. I had a professor that drew a 2D picture on a board where he labeled one axis R^m and the other R^n and then drew a "graph" when discussing something like the implicit function theorem. One takeaway of a lot of linear algebra is that doing this is more-or-less correct (and then functional analysis tells you this is still kind-of correct-ish even in infinite dimensions). Actually SVD tells you in some sense that if you look at things in the right way, the "heart" of a linear map is just 1D multiplication acting independently along different axes, so you don't need to consider all n dimensions at once.