Viewing matrices as manifolds still gives me headaches. Manifolds in the shape of $M \subset \mathbb{R}^n$ I can handle quite well but $M \subset \mathbb{R}^{n\times n}$ is a constraint to deal with In fact I can't depict such higher Manifolds the same way like in normal vector space. For instance what happens to parametrization? Considering a smooth manifolds $M = \{(x,y,z)\in\mathbb{R}^3 \vert \quad x^2+y^2+z^2 = 1\}$ it becomes pretty obvious this is a sphere, parametrized by $\psi(\varphi,\theta) = \left(\begin{array}{c}\cos\varphi\sin(\theta) \\ \sin(\varphi)\,\sin(\theta)\\\cos(\theta)\end{array}\right)$ with $\varphi \in[0,2\,\pi]$ and $\theta \in [0,\pi]$.
But how is that transferable to matrices, like the orthogonal group $M = O(n) = \{A \in\mathbb{R}^{n\times n}\vert \quad A^T\,A = \mathbb{1}\}$.
Imagining each matrix as a point, as suggested in a prior post doesn't help a lot. Maybe because it's just a dimension too much?