How to prove $F_{n-1}F_{n-2}-F_nF_{n-3}=(-1)^n$, where $F_n$ is nth fibonacci number?

Question

I'm struggling to prove the following equation: $$ F_{n-1}F_{n-2}-F_nF_{n-3}=(-1)^n $$ You can verify this fact with this python code.

I managed to simplify the equation a little, so that only $F_{n-2}$ and $F_{n-3}$ left: $$ (F_{n-2})^2-F_{n-2}F_{n-3}-(F_{n-3})^2=(-1)^n $$ But I still don't see how to prove that.

Thanks in advance!