Definition
The subgroup of the general linear group is called the orthogonal group.
The set of matrices with determinant 1 is called the special orthogonal group. (Note that orthogonal matrices all must have determinant .)
Geometrically we can think of the elements of as rotations in -dimensions and are mixtures of rotations and reflections.
Lie group structure
Using the same technique outlined in explanation for why the unitary group is a Lie group, we can see that is a closed subgroup.
Lie algebra
Let be the Lie algebra of and .
Note the Lie algebra of and are the same because all skew-symmetric (real) matrices have trace 0.