TITLE: How to correctly generate uniformly distibuted random elements from SO(n)?
QUESTION [8 upvotes]: I already found some way to produce such matrices from SO(n) with a method called subgroup algorithm but I would like some advice on the method I used. Nowhere I could really find any paper relating to SO(n) directly but rather O(n).
To make it simple, I generate a $(k-1) \times (k-1)$ rotation matrix $\Gamma_{k-1}$, create the matrix 
$\left( \begin{array}{ccc} 1 & \cdots & 0 \\\ \vdots & \Gamma_{n-1} & \\\ 0 & &  \end{array} \right)$
then I find a rotation $R$ that transforms the first column into a vector $v$ generated randomly as a point on the unit $k$-sphere. Then generate a new $k \times k$ rotation matrix $\Gamma_k$ :
$\Gamma_{k} = R\left( \begin{array}{ccc} 1 & \cdots & 0 \\\ \vdots & \Gamma_{k-1} & \\\ 0 & &  \end{array} \right)$
Induction is doing all the work. In practice, either you start by a $1 \times 1$ matrix with single element equal to 1, or you create a first trivial $2 \times 2$ rotation matrix with a angle taken uniformly from $[0,2\pi)$ which seems more direct. I skipped the details on how to generate the matrix $R$ and the vector $v$ as it seemed useless at first. To sum up $v$ is the normalization of a vector $v'$ which elements $v'_i \sim N(0,1)$ and $R$ is obtained by a double Householder reflection.
In practice this seems like a correct incarnation of the subgroup algorithm where the group is SO(n) but I wanted to check with more mathematically inclined than me if it didn't exist a more efficient way! As I said, I found some papers explaining variations or improvements on the original method from Stewart$-$which the subgroup algorithm is a generalization$-$in papers from Anderson or Genz. All of them focused on orthogonal matrices and I would have liked to transpose these improvements to special orthogonal matrices.
See: 

"The Efficient generation of Random Orthogonal Matrices with an Application to Condition Estimators" Stewart
"Generation of Random Orthogonal Matrices" Anderson
"Methods for Generating Random Orthogonal Matrices" Genz
"The Subgroup Algorithm for Generating Uniform Random Variables" Diaconis

EDIT: by the way, I might use this algorithm with n as large as 1000 and beyond
So the real question is (and I'm not really sure how to formulate that correctly as I have a serious lack of knowledge in that area), is there a mapping from O(n) to SO(n) that will preserve the uniformity?

REPLY [3 votes]: The best algorithm is the one due to G.W. Stewart
Stewart, G. W., The efficient generation of random orthogonal matrices with an application to condition estimators. (With mircofiche section), SIAM J. Numer. Anal. 17, 403-409 (1980). ZBL0443.65027.
Algorithm: Let $M$ be $n\times n$ with i.i.d $N(0, 1)$ entries. Let $M=QR$ be the QR factorization ($Q$ orthogonal, $R$ upper triangular). Then $Q$ is uniformly distributed on $O(n)$ (project to $SO(n)$ if you like).