I've been studying differential geometry and algebraic topology for a bit, and something that keeps coming up are the manifolds $GL(n,\mathbb R),$ $SL(n,\mathbb R)$, $O(n),$ $SO(n),$ etc.
I'm wondering if there are any readily available tools to visualize these manifolds as they exist inside of $\mathbb R^{n^2},$ at least for small $n$ (maybe even just $n=2$). Of course, one can get a feeling for what these things look like by studying their topological and geometric properties (homotopy and (co)homology groups, etc.), but I'm interested specifically in how they are embedded into Euclidean space. A Google search returned no results, so I figured I would ask here.
I sort of have in mind a model of $\mathbb R^3$ and a slider so one can see $3$-D cross-sections of these manifolds, but really anything helping to visualize them would be great.