In this paper, the effective use of multiple quadrotor drones as an aerial antenna array that provides wireless service to ground users is investigated. In particular, under the goal of minimizing the airborne service time needed for communicating with ground users, a novel framework for deploying and operating a drone-based antenna array system whose elements are single-antenna drones is proposed. In the considered model, the service time is minimized by minimizing the wireless transmission time as well as the control time that is needed for movement and stabilization of the drones. To minimize the transmission time, first, the antenna array gain is maximized by optimizing the drone spacing within the array. In this case, using perturbation techniques, the drone spacing optimization problem is addressed by solving successive, perturbed convex optimization problems. Then, the optimal locations of the drones around the array's center are derived such that the transmission time for the user is minimized. Given the determined optimal locations of drones, the drones must spend a control time to adjust their positions dynamically so as to serve multiple users. To minimize this control time of the quadrotor drones, the speed of rotors is optimally adjusted based on both the destinations of the drones and external forces (e.g., wind and gravity). In particular, using bang-bang control theory, the optimal rotors' speeds as well as the minimum control time are derived in closed-form. Simulation results show that the proposed approach can significantly reduce the service time to ground users compared to a fixed-array case in which the same number of drones form a fixed uniform antenna array. The results also show that, in comparison with the fixed-array case, the network's spectral efficiency can be improved by 32% while leveraging the drone antenna array system.