A professor told me: "The rotation matrix has 3 unit-norm and 3 orthogonal constraints. Since the matrix is composed of 9 scalars, we have 6 constraints and 3 degrees of freedom (free variables."
In general, if I have $m$ linearly independent constraints, and $N$ variables, in a linear system, then I need $N - m$ solution variables to fill in $N-m$ degrees of freedom / free variables.
But what if these constraints are non-linear, such as the rotation case? I have no idea how to make some general claim about how many degrees of freedom there are for an $N$-variable system with $m$ non-linear constraints.