Rotation matrices and prior invariance for arbitrary dimensions I have a question about a rotation matrix, which can be represented in 2 dimensions as:
$$R_{2}(\theta)=\begin{bmatrix}  \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$$
For some arbitrary angle $\theta$.  This can be extended to an arbitrary number of dimensions by adding an identity matrix:
$$R_{n}(\theta)=\begin{bmatrix} R_{2}(\theta) & 0 \\ 0 & I_{n-2}\end{bmatrix}$$
I have found some "invariance" properties of a n-dimensional prior distribution when rotated in 2 arbitrary dimensions.  My question is: can any rotation in arbitrary dimensions be represented by a sequence of 2-D rotations?  It doesn't matter if the sequence is unique or not for my purposes.
Or perhaps a better question is: if a prior distribution is invariant when rotated about 2 arbitrary dimensions, is it invariant when rotated about an arbitrary number of dimensions?
 A: The answer, I believe, to your first question is "yes". This can be accomplished with Givens rotations, which allow for the annihilation of arbitrary elements of a matrix via a $2\times 2$ rotation matrix. The implication is that if you start with a rotation matrix, then you can reduce it to a diagonal matrix via Givens rotations. But, since orthogonality of a matrix is preserved by multiplication with another orthogonal matrix, this means that the diagonal matrix must be orthogonal, and hence, must contain only 1's and -1's. Additional rotations then reduce this matrix to the identity. The affirmative answer to your first question follows immediately. 
Thus the space of $n\times n$ orthogonal matrices is spanned by Givens rotations with respect to matrix multiplication. 
If this doesn't give you enough detail, let me know and I'll fill it in. 
A: Yes, arbitrary rotations in an $n$ dimensional space can be written as the compositions of Givens rotations. The other question (and I think what you're aiming at) is

Or perhaps a better question is: if a
  prior distribution is invariant when
  rotated about 2 arbitrary dimensions,
  is it invariant when rotated about an
  arbitrary number of dimensions?

The answer is yes, and moreover look no further! since this class of distribution has been completely characterized: it's a subset of the elliptical distributions, when $\Sigma=I$. Elliptical distributions are all and only the rotationally invariant distirbutions, after an affine transformation of its variables. The standard normal multivariate distribution is its best-known example.
