Formats of 6 x 6 skew matrices of linear forms with vanishing Pfaffian

We show that every skew-symmetric 6 × 6 matrix of linear forms with vanishing Pfaffian is congruent to one of finitely many types of matrices, each of which is characterised by a specific pattern of zeroes (and some other linear relations) among its entries. Such matrices are for example important for compactifying moduli spaces of stable rank 2 vector bundles with Chern classes c1 = 0, c2 = 2 on cubic threefolds.