Spiking neural network, consisting of spiking neurons and plastic synapses, is a promising but relatively underdeveloped neural network for neuromorphic computing. Inspired by the human brain, it provides a unique solution for highly efficient data processing. Recently, memristor-based neurons and synapses are becoming intriguing candidates to build spiking neural networks in hardware, owing to the close resemblance between their device dynamics and the biological counterparts. However, the functionalities of memristor-based neurons are currently very limited, and a hardware demonstration of fully memristor-based spiking neural networks supporting in situ learning is very challenging. Here, a hybrid spiking neuron by combining the memristor with simple digital circuits is designed and implemented in hardware to enhance the neuron functions. The hybrid neuron with memristive dynamics not only realizes the basic leaky integrate-and-fire neuron function but also enables the in situ tuning of the connected synaptic weights. Finally, a fully hardware spiking neural network with the hybrid neurons and memristive synapses is experimentally demonstrated for the first time, with which in situ Hebbian learning is achieved. This work opens up a way towards the implementation of spiking neurons, supporting in situ learning for future neuromorphic computing systems.