Special orthogonal group over reals

Definition
Suppose $$n$$ is a natural number. The special orthogonal group of degree $$n$$ over the reals, denoted $$SO(n,\R)$$, is a Lie group that can be defined concretely as the group of $$n \times n$$ matrices with real entries whose determinant is 1 and whose product with the transpose is the identity matrix:

$$SO(n,\R) = \{ A \in \operatorname{Mat}_n(\R) : \operatorname{det} A = 1, \ \operatorname{and} \ A^T = A^{-1} \}$$