Let V be a linear space of even dimension n over a field F of characteristic 0. A subspace W ⊂ ∧ 2 V is maximal singular if rank(w) ≤ n − 1 for all w ∈ W and any W W ⊂ ∧ 2 V contains a nonsingular matrix.It is shown that if W ⊂ ∧ 2 V is a maximal singular subspace which is generated by decomposable elements then dim W ≥ 3n 2 − 3 and that this bound is sharp. The main tool in the proof is the Lovász Matroid Parity Theorem.
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