Synapse emulation is very important for realizing neuromorphic computing, which could overcome the energy and throughput limitations of today's computing architectures. Memristors have been extensively studied for using in nonvolatile memory storage and neuromorphic computing. In this paper, we report the fabrication of vertical sandwiched memristor device using ultrathin quasi-two-dimensional gallium oxide produced by squeegee method. The as-fabricated two-terminal memristor device exhibited the essential functions of biological synapses, such as depression and potentiation of synaptic weight, transition from short time memory to long time memory, spike-timing-dependent plasticity, and spike-rate-dependent plasticity. The synaptic weight of the memristor could be tuned by the applied voltage pulse, number, width, and frequency. We believe that the injection of the top Ag cations should play a significant role for the memristor phenomenon. The ultrathin of medium layer represents an advance to integration in vertical direction for future applications and our results provide an alternative way to fabricate synaptic devices.