[mathpepe]

In the case of matrices with real coefficients, these matrices constitute the famous ortogonal group. The ortogonal group is a Lie group of dimension n*(n-1)/2. A table of Lie groups with their dimensions is here https://en.wikipedia.org/wiki/Table_of_Lie_groups

Every Lie group has an associated vector space with the same dimension, so many question about Lie groups can be addressed by studying the associate vector space.

To suggest an application of Lie groups and their associated vector spaces, in machine learning one can reduce the number of parameters when a group act over a space (rotations and the like), so you obtain new features to train your machine in a more efficient way.