Generalized rotation matrix in N dimensional space around N-2 unit vector

Solution 1:

The definition is that $A\in M_{n}(\mathbb{R})$ is called a rotation matrix if there exist a unitary matrix $P$ s.t $P^{-1}AP$ is of the form $$\begin{pmatrix}\cos(\theta) &-\sin(\theta)\\ \sin(\theta) & \cos(\theta)\\ & & 1\\ & & & 1\\ & & & & 1\\ & & & & & .\\ & & & & & & .\\ & & & & & & & .\\ & & & & & & & & 1 \end{pmatrix}$$

If we consider $A:\mathbb{R}^{n}\to\mathbb{R}^{n}$ then the meaning is that there exist an orthonormal basis where we rotate the $2-$dimensional space spanned by the first two vectors by angle $\theta$ and we fix the other $n-2$ dimensions