Generally, the applications of Wireless Sensor Networks (WSNs) need a deployment of a large population of sensor nodes. Therefore, the design of such networks requires scalable organizational and management strategies. To ensure network scalability and energy saving, sensor nodes could be grouped into clusters. WSNs' research community has given a high importance to clustering techniques in these networks during the last years. However, most of the existing clustering schemes focus on making sensor nodes periodically taking turns in fulfilling the burdening task of cluster heads. These solutions do not consider the consumed energy by the clustering mechanism itself which leads to waste the energies of sensor nodes. To cope with this problem, we propose an energy aware clustering scheme for WSNs based on a self-stabilizing algorithm. We term our scheme SEAC (Saving Energy by an Alliance-based Clustering). Sensor nodes execute a distributed algorithm which determines the set of nodes that form an alliance based on an energy condition. The alliance's sensor nodes are the cluster heads. SEAC does not perform re-clustering at each round. New cluster heads are selected when the energy condition is not verified. The frequency of re-clustering is decreased, which diminishes the communication overhead. Simulation results show that SEAC improves the network lifetime and consumes less energy with respect to existing approaches.