Eric Weisstein's Sphere Point Picking points out that sampling uniformly from each angle $\phi$ and $\theta$ in spherical coordinates does not sample from the uniform sphere because it clusters near the poles. I am interested in which distribution over the angles does sample uniformly over the area element.
For the spherical case, he notes that the random variables $\phi$ and $\theta$ that do correspond to sampling from the uniform sphere are:
$\theta = 2\pi u \\ \phi = \cos^{-1}(2v -1)$
where $u$ and $v$ are random variables uniformly distributed over [0, 1].
I would like to know how this extends to n-dimensional hyperspheres. Is there a similar expression for the distribution of the angles $\boldsymbol{\theta}$ when sampling from a uniform hypersphere?
Very grateful for any help!
(I'm aware that there are simpler ways to sample from the unit hypersphere such as this. I'm specifically interested in the probability density function of the angles.)