Per title ...
There's a question about this on Quora, but unfortunately both proofs are way over my head, and the answers differ as well - working through the expression in the second answer gives the volume of $SO(3)$ as $8\pi^2$, which is different from $2^{9/2} \pi^2$ as in the first answer.
Proposition 2.1 of this paper gives an expression for the volume of $SO(n)$ as well, but unfortunately it's also beyond my level of understanding.