In this paper, we develop a socially cooperative optimal control framework for connected and automated vehicles (CAVs) in mixed traffic using social value orientation (SVO). In our approach, we formulate the interaction between a CAV and a human-driven vehicle (HDV) as a simultaneous game to facilitate the derivation of a Nash equilibrium. In the imposed game, each vehicle minimizes a weighted sum of its egoistic objective and a cooperative objective. The SVO angles are used to quantify preferences of the vehicles toward the egoistic and cooperative objectives which lead to an appropriate design of weighting factors in a multi-objective optimal control problem. We prove that by solving the proposed optimal control problem, a Nash equilibrium can be obtained. To estimate the SVO angle of the HDV, we develop a receding horizon estimation based on maximum entropy inverse reinforcement learning. The effectiveness of the proposed approach is demonstrated by numerical simulations at a highway on-ramp merging scenario.