On bilinear maps determined by rank one idempotents

Let Mn, n⩾2, be the algebra of all n×n matrices over a field F of characteristic not 2, and let Φ be a bilinear map from Mn×Mn into an arbitrary vector space X over F. Our main result states that if ϕ(e,f)=0 whenever e and f are orthogonal rank one idempotents, then there exist linear maps Φ1,Φ2:Mn→X such that ϕ(a,b)=Φ1(ab)+Φ2(ba) for all a,b∈Mn. This is applicable to some linear preserver problems.