Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)The concept of matrix of rotation generalizes the rotation number for stochastic dynamical systems given in [11]. This matrix is the asymptotic time average of the Maurer-Cartan form composed with the Riemannian connection along the induced trajectory in the orthonormal frame bundle OM over an n-dimensional Riemannian manifold M. It provides the asymptotic behaviour of an orthonormal n-frame under the action of the derivative flow and the Gram-Schmidt orthonormalization. We lift the stochastic differential equation of the system on M to a stochastic differential equation in OM and we use Furstenberg-Khasminskii argument to prove that the matrix of rotation exists almost surely with respect to invariant measures on this bundle.182231245Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)