This paper shows that the aerodynamic effects can be compensated in a quadrotor system by means of a control allocation approach using neural networks. Thus, the system performance can be improved by replacing the classic allocation matrix, without using the aerodynamic inflow equations directly. The network training is performed offline, which requires low computational power. The target system is a Parrot MAMBO drone whose flight control is composed of PD-PID controllers followed by the proposed neural network control allocation algorithm. Such a quadrotor is particularly susceptible to the aerodynamics effects of interest to this work, because of its small size. We compared the mechanical torques commanded by the flight controller, i.e., the control input, to those actually generated by the actuators and established at the aircraft. It was observed that the proposed neural network was able to closely match them, while the classic allocation matrix could not achieve that. The allocation error was also determined in both cases. Furthermore, the closed-loop performance also improved with the use of the proposed neural network control allocation, as well as the quality of the thrust and torque signals, in which we perceived a much less noisy behavior.