  Random orthogonal matrices play an important role in probability and
statistics, arising in multivariate analysis, directional statistics, and
models of physical systems, among other areas. Calculations involving random
orthogonal matrices are complicated by their constrained support. Accordingly,
we parametrize the Stiefel and Grassmann manifolds, represented as subsets of
orthogonal matrices, in terms of Euclidean parameters using the Cayley
transform. We derive the necessary Jacobian terms for change of variables
formulas. Given a density defined on the Stiefel or Grassmann manifold, these
allow us to specify the corresponding density for the Euclidean parameters, and
vice versa. As an application, we describe and illustrate through examples a
Markov chain Monte Carlo approach to simulating from distributions on the
Stiefel and Grassmann manifolds. Finally, we establish an asymptotic
independent normal approximation for the distribution of the Euclidean
parameters which corresponds to the uniform distribution on the Stiefel
manifold. This result contributes to the growing literature on normal
approximations to the entries of random orthogonal matrices or transformations
thereof.
