  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.
