Given n-dimensional orthonormal basis with positive orientation, can I reach the standard basis via a sequence of rotations?

Question

Given a n-dimensional orthonormal basis in Euclidean space with positive orientation, can I reach the standard basis via a sequence of rotations?

Answer 1

Yes. The matrix going between these two bases is in the special orthogonal group and is conjugate there to a matrix with block decomposition $$\pmatrix{R_{t_1}&&&\\&R_{t_2}&&\\&&\ddots&\\&&&R_{t_m}}$$ or $$\pmatrix{R_{t_1}&&&&\\&R_{t_2}&&&\\&&\ddots&&\\&&&R_{t_m}\\&&&&1}$$ where $R_t$ is a rotation matrix in two dimensions.