Let $V$ be a vector space with the orthonormal basis $Q = \{ \vec{q_1},\ldots, \vec{q_n} \}$ and let $\ell:V\to V$ be an orthogonal map. Prove that the matrix $L$ of of $\ell$ with respect to $Q$ is orthogonal.
Note: By orthogonal map I mean that $\ell$ is linear and satisfies $\left\Vert \ell(\vec{x}) \right\Vert = \left\Vert \vec{x} \right\Vert$ for all $x \in V$. By orthogonal matrix I mean that $L$ has orthonormal columns.
I have that $ L=\left[ \begin{array}{ccc} [\ell(\vec{q_1})]_Q & \cdots & [\ell(\vec{q_n})]_Q \end{array} \right] $
but I don't have any idea how to proceed.