Why does an $n \times n$ rotation matrix have $\frac{1}{2}n(n-1)$ undetermined parameters? Consider an orthogonal transformation between Cartesian coordinate systems in $n$-dimensional space. The $n \times n$ rotation matrix 
$$R = \left(a_{ij}\right)$$ 
has $n^2$ entries. These are not independent; they are related by the orthogonality conditions
$$a_{ik}a_{jk} = \delta_{ij},$$ 
which are $\frac{1}{2}\!\!\left(n^2 + n\right)$ independent equations in the $a_{ij}$. Thus, 
$$n^2 - \frac{1}{2}\!\!\left(n^2 + n\right) = \frac{1}{2}n\left(n-1 \right)$$
of the $a_{ij}$ are left undetermined. 
Why is the last step justified? If there are $n$ independent equations, is it always possible to solve them for $n$ unknowns (even if the equations involve the sums of quadratic terms, as they do here)? If so, why? Under what general conditions is it possible to solve a system of $n$ independent equations for $n$ unknowns? 

If you want something a bit more concrete, pick one row at a time.  The first row is any norm 1 vector, so that is $n-1$ parameters.  The second row is any norm 1 vector perpendicular to the first row, so that is $n-2$ parameters.  The $k$ row is any norm 1 vector perpendicular to the first $k-1$ columns, so that is $n-k$ parameters.  You end up with $(n-1) + (n-2) + \cdots + 2 + 1 + 0 = \frac12 n(n-1)$ parameters.
