The background of this question is how a random variable $X$ on the orthogonal group $O(n)$ whose distribution is the normalized Haar measure $\mu$, i.e., $\mu( O(n) ) = 1$, can be realized on a computer that has access to a number of "simpler" random variables, like:


* a perfect coin
* the uniform probability measure on $[0,1]$ or $S^1$
* Gaussian variables on $\mathbb R$ with arbitrary standard deviation.


An algorithm needs to use the random variable $X$, but I am not aware of an explicit formula that would allow for such a reduction.



Haar measure on $O(n)$ reduced to simpler probability space
Starting from a real $N\times N$ matrix and essentially performing a $QR$ decomposition, then if the initial real matrix elements are independent identically distributed Gaussian, then the matrices $Q$ will be Haar-distributed on $O(N)$. Please see the full construction in the following [article](http://www.maths.bris.ac.uk/~mafm/bibliography/art07/rand_mat.pdf) by: Francesco Mezzadri


