A decomposition result for the Haar distribution on the orthogonal group
  Let H be a Haar distributed random matrix on the group of pxp real orthogonal matrices. Partition H into four blocks: (1) the (1,1) element, (2)the rest of the first row, (3) the rest of the first column, and (4)the remaining (p-1)x(p-1) matrix. The marginal distribution of (1) is well known. In this paper, we give the conditional distribution of (2) and (3) given (1), and the conditional distribution of (4) given (1), (2), (3). This conditional specification uniquely determines the Haar distribution. The two conditional distributions involve well known probability distributions namely, the uniform distribution on the unit sphere in p-1 dimensional space and the Haar distribution on (p-2)x(p-2) orthogonal matrices. Our results show how to construct the Haar distribution on pxp orthogonal matrices from the Haar distribution on (p-2)x(p-2) orthogonal matrices coupled with the uniform distribution on the unit sphere in p-1 dimensions. 
Morris L. Eaton and Robb J. Muirhead