Special orthogonal group over reals

Definition
For any natural number $$n$$, the special orthogonal group over reals of degree $$n$$, denoted $$SO(n,\R)$$ or $$SO_n(\R)$$, is defined as the following group:

$$\{ A \in GL(n,\R) \mid \det(A) = 1, AA^T = I \}$$

This can also be described as the group of linear transformations of $$\R^n$$ that are orientation-preserving and also preserve the dot product.