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Spiking neural network (SNN) has emerged as one of the popular architectures in complex pattern recognition and classification tasks. However, hardware implementation of such algorithms using conventional CMOS based neuron consume resources and power that are orders of magnitude higher than that in human brain. This can be attributed to the mismatch of the computational architecture between biological brain and the current Boolean logic computing platform. Magnetic skyrmions have been intensively studied as a prospective information carrier in neuromorphic computing hardware design. In this work, a compact time-domain skyrmion-racing winner-takes-all (SR-WTA) leaky-integrate-fire (LIF) spiking neuron network is presented for the first time. The skyrmion motion dynamics in the LIF neuron and the behaviors of the neuron network was investigated comprehensively. Both SPICE and micromagnetic simulations are performed to evaluate the functionality and performance of the proposed SR-WTA based SNN.