  We show that every skew-symmetric 6 x 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
c_1=0, c_2=2 on cubic threefolds.
