How to determine the rank of a matrix if on the diagonal it's all zeros?

Question

Can someone tell, what's the way to determine the rank for a quadratic form? I know one way to determine the signature, that is by finding the eigenvalues. For a quadratic form $f=x_1x_2-x_2x_3$ the signature would be $(1,1)$, because there is one positive and one negative eigenvalue. But how do I determine the rank? I know that you can complete the square and then determine. I am not sure whether I do the completion of the squares the right way: $(x_1+\frac{1}{2}x_2)^2-\frac{1}{4}x_2^2-x_1^2+(x_2-\frac{1}{2}x_3)^2-x_2^2-\frac{1}{4}x_3^2$ I am quite sure that this is no the right way for completing the square, can someone explain how can I do it properly and also how can I determine the rank after completion of the square?