Uniform parametrisation of $SO(3)$

Question

Does there exist a map from a hypercube of some dimension to $SO(3)$ such that pushforward measure of standard measure on the hypercube is invariant under action of $SO(3)$ on itself?

Answer 1

There is a very general theorem that takes care of this (but maybe too general for your purposes), it is a generalization of Moser's Theorem:

Suppose that $M$ is an $m$-dimensional compact manifold with corners (say, a cube), equipped with two volume forms $\omega_1, \omega_2$ of equal volume. Then there is a diffeomorphism $f: M\to M$ such that $f^*(\omega_2)=\omega_1$.

See

Bruveris, Martins; Michor, Peter W.; Parusiński, Adam; Rainer, Armin, Moser’s theorem on manifolds with corners, Proc. Am. Math. Soc. 146, No. 11, 4889-4897 (2018). ZBL1398.58004.

Now, in your situation, take $G=SO(3)$; normalize its volume form to have unit volume. Using for instance Euler angles obtain a parameterization of $G$ by the 3-dimensional unit cube $I^3$. Pull-back the volume form from $G$ to a form $\omega_2$ on $I^3$. Let $\omega_1$ be the standard volume form on $I^3$. Then, generalized Moser's theorem will imply existence of a diffeomorphism $f: I^3\to I^3$ such that $f^*(\omega_2)=\omega_1$, i.e. $f_*(\omega_1)=\omega_2$. Now, compose with the above parameterization $I^3\to G$.