I posted this on MSE, but no answer is received, so I post this here.

The problem of finding the $n(n-1)/2$-dimensional volume of the set $SO(n)\subset\mathbb R^{n^2}$ is asked before in this MO post. In the comment section, it is suggested that the answer is $\prod^{n-1}_{k=1}2^{(k-1)/2}\cdot \textrm{volume}(S^k)$ (where $S^k$ denotes the unit $k$-sphere). It was pointed out that, as I quote:

"... the factor of $2^{(k-1)/2}$ that you have to put in at each level because the natural map $\pi:SO(k)→S^{k−1}$ is not a Riemannian submersion, but is a Riemannian submersion scaled by a factor of $1/{\sqrt 2}$, i.e., each horizontal vector for this bundle is shrunk by a factor of $\sqrt 2$ by the differential of $\pi$."

My question is:

Explicitly, what is the map $\pi$? How to use $\pi$ to calculate the volume of $SO(n)$? How to compute $d\pi$? Thanks.