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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.

Allure
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    You have to specify what metric you're putting on $\text{SO}(n)$. It's kind of like asking, "What is the volume of a sphere?" It depends: Do you mean the sphere of radius $1$ or the sphere of radius $17$ or ....? – Jesse Madnick Mar 27 '20 at 08:19
  • Indeed it should be specified. The Killing form provides a natural normalization. Sometimes one requires that the max of the sectional curvature is $+1$, which yields another normalization. – YCor Mar 27 '20 at 22:40
  • @JesseMadnick hmm I don't know what the metric is for my application, but I do know the result I expect for SO(3). Does this mean I can safely use the answer on Quora that gives that result and extent to SO(n)? – Allure Mar 28 '20 at 08:50

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