Let $\Lambda$ be an unimodular lattice with a quadratic form $(-,-)$ of signature $(m,n)$ , $m,n>0$.

I know that, fixed a base $e_1,\cdots,e_{m+n}$ for $\Lambda$, the matrix which has entries $a_{i,j}=(e_i,e_j)$ has determinant ugual to $\pm 1$.

I wonder if in this case it is always possibile to find an orthogonal base, i.e. a base $f_1,\cdots,f_{m+n}$ with $(f_i,f_i)=\pm 1$ and $(f_i,f_j)=0$.

I think yes, but i can't really prove it.