For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix
$$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
Show that for $X \in \mathbb{R}^{3}$, $\mathcal{M}X \cdot X = 0$. Geometrically interpret the application $\mathbb{R}^{3} \rightarrow \mathbb{R}^{3},\, X \mapsto (I_{3} + \mathcal{M})X$.