The bilinear forms graph denoted here by Bilq(d×e) is a graph defined on the set of (d×e)-matrices (e≥d) over $$\mathbb{F}_q$$ with two matrices being adjacent if and only if the rank of their difference equals 1. In 1999, K. Metsch showed that the bilinear forms graph Bilq(d×e), d≥3, is characterized by its intersection array if one of the following holds: — q=2 and e≥d+4 — q≥3 and e≥d+3. Thus, the following cases have been left unsettled: — q=2 and e∈{d,d+1,d+2,d+3} — q≥3 and e∈{d,d+1,d+2}. In this work, we show that the graph of bilinear (d×d)-forms over the binary field, where d≥3, is characterized by its intersection array. In doing so, we also classify locally grid graphs whose μ-graphs are hexagons and their intersection numbers bi,ci are well-defined for all i=0,1,2.