Estimation of User Terminals' (UTs') Angle of Arrival (AoA) plays a significant role in the next generation of wireless systems. Due to high demands, energy efficiency concerns, and scarcity of available resources, it is pivotal how these resources are used. Installed antennas and their corresponding hardware at the Base Station (BS) are of these resources. In this paper, we address the problem of antenna selection to minimize Cramer-Rao Lower Bound (CRLB) of a planar antenna array when fewer antennas than total available antennas have to be used for a UT. First, the optimal antenna selection strategy to minimize the expected CRLB is proposed. Then, using this strategy as a preliminary step, we present a two-stage greedy antenna selection method whose goal is to minimize the instantaneous CRLB. The optimal start point of the greedy algorithm is presented alongside some methods to reduce the algorithm's computational complexity. Numerical results confirm the accuracy of proposed solutions. They demonstrate that the proposed antenna selection method only requires a small proportion of the total available antennas to accomplish a significant amount of the total performance, enhancing hardware utilization efficiency. Also, it is shown that the presented algorithm has a high error tolerance.