We theoretically show that moderate interaction between electrons confined to move in a plane and localized magnetic moments leads to formation of a noncoplanar magnetic state. The state is similar to the Skyrmion crystal recently observed in cubic systems with the Dzyaloshinskii-Moriya interaction; however, it does not require spin-orbit interaction. The noncoplanar magnetism is accompanied by the ground-state electrical and spin currents, generated via the real-space Berry phase mechanism. We examine the stability of the state with respect to lattice discreteness effects and the magnitude of magnetic exchange interaction. The state can be realized in a number of transition metal and magnetic semiconductor systems.