TITLE: How to count the number of free parameters in an orthogonal transformation matrix?
QUESTION [7 upvotes]: 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  
$$ M_2 = \pm\left(\begin{array}{cc}
\cos\theta & \sin\theta\\
-\sin\theta & \cos\theta
\end{array}\right) .$$

REPLY [5 votes]: Use the dictionary "orthogonal transformation $\leftrightarrow$ orthonormal basis". For the first basis vector, choose any point on the unit sphere ($n-1$ free parameters). For the second basis vector, choose any point on the unit sphere which is orthogonal to the first basis vector ($n-2$ free parameters). Etc