With increasing applied current we show that the moving vortex lattice changes its structure from a triangular one to a set of parallel vortex rows in a pinning free superconductor. This effect originates from the change of the shape of the vortex core due to non-equilibrium effects (similar to the mechanism of vortex motion instability in the Larkin-Ovchinnikov theory). The moving vortex creates a deficit of quasiparticles in front of its motion and an excess of quasiparticles behind the core of the moving vortex. This results in the appearance of a wake (region with suppressed order parameter) behind the vortex which attracts other vortices resulting in an effective direction-dependent interaction between vortices. When the vortex velocity \(v\) reaches the critical value \(v_c\) quasi-phase slip lines (lines with fast vortex motion) appear which may coexist with slowly moving vortices between such lines. Our results are found within the framework of the time-dependent Ginzburg-Landau equations and are strictly valid when the coherence length \(\xi(T)\) is larger or comparable with the decay length \(L_{in}\) of the non-equilibrium quasiparticle distribution function. We qualitatively explain experiments on the instability of vortex flow at low magnetic fields when the distance between vortices \(a \gg L_{in} \gg \xi (T)\). We speculate that a similar instability of the vortex lattice should exist for \(v>v_c\) even when \(a<L_{in}\).