Prove that every bilinear form $f:\mathbb R^n \times \mathbb R^n\rightarrow \mathbb R$ has a basis $\{v_1,\ldots,v_n\} \subset \mathbb R^n$ such that $f(v_i,v_j)=-f(v_j,v_i)$ for every $i\neq j$.
I have no idea how to start thinking about this one. Should I actually "build" the required basis or what? Any ideas ?
thanks