In this paper, we study cokernels of random $n\times n$ matrices over $\mathbb Z$ with symmetry conditions determined by fixed alternating bilinear forms on $\mathbb Z^n$. These include perturbations of random symmetric matrices at a very small (but unbounded with $n$) number of entries. We show that, subject to fairly weak conditions on the distributions of the entries, the distribution of these cokernels converges weakly to the Cohen-Lenstra distribution, which is the limiting distribution of cokernels of random matrices with no symmetry constraints. This result demonstrates that the cokernel distributions of symmetric matrices are quite sensitive to small perturbations of the symmetry conditions.