Hurwitz defined a decomposition of the Haar measure on $SO(n)$ based on Given's rotation. So by left multiplication of Givens rotation one can always bring an orthogonal matrix into the identity. The order of multiplication is $R(1,2;\theta_1)R(2,3;\theta_2) \ldots R(n-1,n;\theta_{n-1}) R(2,3;\theta_n) \ldots R(n-1,n;\theta_{\binom{n}{2}})$ where the $\theta$'s are independent and have certain beta distributions (not identically distributed). My question is whether this decomposition holds for other product of Givens rotations. Also if we take uniform iid $\theta_j$'s, is the singularity always on the set where some $\theta_j = 0$ or $\pi$? This is the case for the Hurwitz decomposition. If not please describe the singular set for iid uniform $\theta_j$'s. For a reference of Hurwitz decomposition, see the paper by Diaconis and Saloff-Coste: www-stat.stanford.edu/~cgates/PERSI/papers/kac10.pdf