Orthogonal Group
Definition
$n \times n$ The set of orthogonal matrices is denoted by $\mathrm{O}(n)$ and is called the $n$-dimensional orthogonal group.
$$ \mathrm{O}(n) := {\left\{ A \in M_{n \times n}(\mathbb{R}) : AA^{T} = I \right\}} $$
Description
Since it is a set of orthogonal matrices, only invertible matrices exist. Hence, it forms a group with respect to matrix multiplication, and is a subgroup of the general linear group $\mathrm{GL}(n, \mathbb{R})$.
It has a differentiable structure, thereby constituting a Lie group.