SO(n) as a manifold

Question

I cannot find some basic information on $SO(n)$ ($n$ general, not just 3) as a manifold: what is the geodesic distance between two matrices, what are the eigenfunctions and eigenvalues of the Laplace-Beltrami operator. It would be good to get a recursive definition starting from $SO(2)$, where the answers are trivial.

Answer 1

The article On strain measures and the geodesic distance to $SO_n$ in the general linear group has an introductory section on the geodesic distance, which might be helpful. In general I would look into Differential Geometry, Lie Groups, and Symmetric Spaces by Sigurdur Helgason.