Emotion-cause pair extraction (ECPE) is a new task which aims at extracting the potential clause pairs of emotions and corresponding causes in a document. To tackle this task, a two-step method was proposed by previous study which first extracted emotion clauses and cause clauses individually, then paired the emotion and cause clauses, and filtered out the pairs without causality. Different from this method that separated the detection and the matching of emotion and cause into two steps, we propose a Symmetric Local Search Network (SLSN) model to perform the detection and matching simultaneously by local search. SLSN consists of two symmetric subnetworks, namely the emotion subnetwork and the cause subnetwork. Each subnetwork is composed of a clause representation learner and a local pair searcher. The local pair searcher is a specially-designed cross-subnetwork component which can extract the local emotion-cause pairs. Experimental results on the ECPE corpus demonstrate the superiority of our SLSN over existing state-of-the-art methods.