A theorem of Bogolyubov states that for every dense set A A in Z N \mathbb {Z}_N we may find a large Bohr set inside A + A − A − A A+A-A-A . In this note, motivated by work on a quantitative inverse theorem for the Gowers U 4 U^4 norm, we prove a bilinear variant of this result for vector spaces over finite fields. Given a subset A ⊂ F p n × F p n A \subset \mathbb {F}^n_p \times \mathbb {F}^n_p , we consider two operations: one of them replaces each row of A A by the set difference of it with itself, and the other does the same for columns. We prove that if A A has positive density and these operations are repeated several times, then the resulting set contains a bilinear analogue of a Bohr set, namely the zero set of a biaffine map from F p n × F p n \mathbb {F}^n_p \times \mathbb {F}^n_p to an F p \mathbb {F}_p -vector space of bounded dimension. An almost identical result was proved independently by Bienvenu and Lê.