Spiking neural networks (SNNs) are bio-plausible computing models with high energy efficiency. The temporal dynamics of neurons and synapses enable them to detect temporal patterns and generate sequences. While Backpropagation Through Time (BPTT) is traditionally used to train SNNs, it is not suitable for online learning of embedded applications due to its high computation and memory cost as well as extended latency. Previous works have proposed online learning algorithms, but they often utilize highly simplified spiking neuron models without synaptic dynamics and reset feedback, resulting in subpar performance. In this work, we present Spatiotemporal Online Learning for Synaptic Adaptation (SOLSA), specifically designed for online learning of SNNs composed of Leaky Integrate and Fire (LIF) neurons with exponentially decayed synapses and soft reset. The algorithm not only learns the synaptic weight but also adapts the temporal filters associated to the synapses. Compared to the BPTT algorithm, SOLSA has much lower memory requirement and achieves a more balanced temporal workload distribution. Moreover, SOLSA incorporates enhancement techniques such as scheduled weight update, early stop training and adaptive synapse filter, which speed up the convergence and enhance the learning performance. When compared to other non-BPTT based SNN learning, SOLSA demonstrates an average learning accuracy improvement of 14.2%. Furthermore, compared to BPTT, SOLSA achieves a 5% higher average learning accuracy with a 72% reduction in memory cost.