When a low density brush of single-stranded DNA (ssDNA) targets end-grafted to a surface is immersed in a solution of complementary ssDNA probes, a regular brush of DNA duplexes is formed by 1:1 hybridization between probe and target DNA. We suggest that in higher density brushes of ssDNA this process competes with cross-hybridization of a target strand to several neighboring probe strands resulting in the formation of a cross-linked DNA network. We analyze a simple 2D model of a dense DNA brush and use analytic methods and computer simulations to find how the conditions for network formation depend on system size and DNA length. We argue that in 3D brushes cross-hybridization will nearly always lead to network formation and suggest that this may explain some intriguing results on dense DNA brushes. Experiments on DNA monolayers and concentrated DNA solutions that could test our predictions are proposed.