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Magnetic skyrmions have attracted considerable interest, especially after their recent experimental demonstration at room temperature in multilayers. The robustness, nanoscale size and non-volatility of skyrmions have triggered a substantial amount of research on skyrmion-based low-power, ultra-dense nanocomputing and neuromorphic systems such as artificial synapses. Room-temperature operation is required to integrate skyrmionic synapses in practical future devices. Here, we numerically propose a nanoscale skyrmionic synapse composed of magnetic multilayers that enables room-temperature device operation tailored for optimal synaptic resolution. We demonstrate that when embedding such multilayer skyrmionic synapses in a simple spiking neural network (SNN) with unsupervised learning via the spike-timing-dependent plasticity rule, we can achieve only a 78% classification accuracy in the MNIST handwritten data set under realistic conditions. We propose that this performance can be significantly improved to about 98.61% by using a deep SNN with supervised learning. Our results illustrate that the proposed skyrmionic synapse can be a potential candidate for future energy-efficient neuromorphic edge computing.