Every real matrix $A$ can be factorized $A = Q R$ where $Q$ is orthogonal and $R$ is a (say, upper) triangular matrix (wikipedia/QR decomposition which is a special case of Iwasawa decomposition for semisimple Lie groups). This is consequence of Gram-Schmidt orthogonalization. Similarly, every complex matrix can be factorized into a unitary and a complex upper triangular (complex) matrix.

Over the reals, the Cayley transform is a diffeomorphism between the linear space skewsymmetric matrices and an open subset of the Lie group of orthogonal matrices ($A$ such that ${\operatorname{Id}}+A$ is invertible) - a chart which is often having an advantage over using the exponential map.