Let $\mathbf{n} \in \mathbb{R}^{3}$ be a unit vector. Consider the operation

$$\mathbf{x} \mapsto \mathbf{n} \times \mathbf{x}$$

Write this in matrix form, i.e., find a $3 \times 3$ matrix $\mathbf{A}$ such that $\mathbf{A} \mathbf{x}=\mathbf{n} \times \mathbf{x}$ for all $\mathbf{x}$, and compute the eigenvalues of $\mathbf{A}$. In the case when $\mathbf{n}=(0,0,1)$, compute $\mathbf{A}^{2}$ and its eigenvalues and eigenvectors.