Prove that the number of free parameters in an $n\times n$ orthogonal transformation matrix is equal to $\frac{n(n-1)}{2}$. For example parametrization of $2 \times 2$ orthogonal matrix requires only one parameter, ie $\theta$.
And the parametric form is
$ M_2 = \pm\left(\begin{array}{cc} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{array}\right) .$