Special orthogonal group

Let $\langle\cdot,\cdot\rangle$ be the standard inner product on $\mathbb R^n$. The special orthogonal group is

It is a closed Lie subgroup of the orthogonal group, hence a Lie group (see closed subgroup). For $n\ge 2$, $SO(n)$ is connected, and for $n\ge 3$ it is not simply connected; its universal cover is the spin group.

More generally, one defines $SO(p,q)$ as the determinant-1 subgroup of the group preserving a nondegenerate bilinear form of signature $(p,q)$; these are basic noncompact matrix Lie groups (compare Lorentz group for the $(n-1,1)$ case).

The Lie algebra of $SO(n)$ is the orthogonal Lie algebra $\mathfrak{so}(n)$, consisting of skew-symmetric matrices:

with bracket given by commutator. The group $SO(n)$ is compact, making it a key example in compact Lie group theory.