Say I need to generate a uniformly distributed collection $\{X\}$ of elements of $SU(3)$. (For concreteness, I will consider the distribution to be uniform if the distribution of $X$ is identical to the distribution of $gX$ where $g$ can be any element of $SU(3)$.)
I know how to generate uniform random elements of $SU(2)$ because it is isomorphic with $SO(3)$ and I know how to generate random rotations. In past work, I (and other Lattice Gauge Theory people doing Monte-Carlo simulations) needed to generate random $SU(3)$ elements such that the probability of an element is the same as that of its inverse, and such that the probability density near at $SU(3)$ element is non-zero. However, these are weaker conditions than uniformity, and what we did was to take the product of three elements in three different $SU(2)$ subspaces of $SU(3)$. We could get arbitrarily close to a uniform distribution by multiplying several such products.
But I would like to know how to generate $SU(3)$ elements with uniform distribution, not just almost-uniform distribution.