Let $a_1,\ldots,a_{2n+1}$ be elements of a group $G$. Then $a_1\cdots a_{2n+1}a_1^{-1}\cdots a_{2n+1}^{-1}$ is a product of $n$ commutators.
The case $n=1$ was proven in this question and the proof of the above statement should use that and induction. I'm ashamed to admit that again I have no clue. It's probably quite analogous, but I really can't think of anything but blind guessing.
So it would be nice if someone could help me out.