Compute the determinant of the $\displaystyle 2n \times 2n$ matrix $\displaystyle \left(\begin{array} {ccccc} 0 & 1 & 1 & ... & 1 \\ -1 & 0 & 1 & ... & 1 \\ -1 & -1 & 0 & ... & 1 \\ .. \\ -1 & -1 & ... & -1 & 0 \end{array}\right)$

(ie. all diagonal entries equal to zero, all above diagonal entries equal to 1, all below diagonal entries equal to -1)

I am not sure how to go about finding this in general. I have found that the determinants when $\displaystyle n$ is 1 and 2 are both 1, but that doesn't really help me in general, and I don't really have any idea what to start with for the general case. Any tips?