Let $A$ be an $n\times n$ matrix and $B=\set{\vectorlist{x}{n}}$ be an orthonormal basis of $\complex{n}$. Define \begin{align*} C&=\set{A\vect{x}_1,\,A\vect{x}_2,\,A\vect{x}_3,\,\dots,\,A\vect{x}_n} \end{align*}

Then $A$ is a unitary matrix if and only if $C$ is an orthonormal basis of $\complex{n}$.