Question about definition of rotation on sphere

Question

I just started M.A. and I do not have enough knowledge about aforementioned concepts. I truly want to know everything about this mapping in details. A rotation of the sphere $s^2$ is a map $ r= r_{p,\alpha}$ described by spinning the sphere (actually, spinning the ambient space $R^3$) about the line through the origin and the point $ p\in s^2$, counterclockwise through angle $\alpha $ looking at $p$ from outside the sphere. Thus r is the linear map that fixes $p$ and rotates planes orthogonal to $p$ through angle $\alpha$. Let $q$ be a unit vector orthogonal to $p$. then the matrix of r is (viewing p and q as column vectors) $m_r= \begin{bmatrix} p&q&p\times q \end{bmatrix} \begin{bmatrix} 1&0&0\\ 0&cos\alpha&-sin\alpha\\ 0&sin\alpha&cos\alpha \end{bmatrix} \begin{bmatrix} p&q&p\times q \end{bmatrix}^{-1} $ please help me to know it.