Plane isometry/C/Special/Explicit form/Exercise

Let $M$ be a complex $2\times 2$-matrix such that the columns form an orthonormal basis of ${ℂ}^2$ and such that the determinant equals $1$. Show that $M$ has the form

$M=\left(\begin{array}{cc}u& -\bar{v}\\ v& \bar{u}\end{array}\right),$

with $u,v\in ℂ$ and $\Vert \left(\begin{array}{c}u\\ v\end{array}\right)\Vert =1$.

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