Definition

We define the set of all rotation matrices as the special orthogonal group on $\mathbb{R}^{3}$ $$SO(3):=\{ \mathbf{C}\in\mathbb{R}^{3\times3}\mid \mathbf{CC}^{\top}=\mathbf{1}\,\&\,\det(\mathbf{C})=1 \}$$ i.e. it is the set of all orthonormal matrices on $\mathbb{R}^{3\times3}$.