Koopman operator approximation is becoming a leading trend for the identification and control of nonlinear systems, particularly with the use of Deep Neural Networks (DNNs). Although DNNs have shown potential to simultaneously identify Koopman observation functions and its lifted control dynamics, training these components jointly often leads to reduced model accuracy and robustness. This study proposes a novel learning approach called system-oriented DNN (soDNN), which improves the learning of Koopman observation functions by offering gradient information to update the lifted bilinear system dynamics. Unlike conventional learning strategies, soDNN achieves enhanced model precision for both short- and long-term predictions, as demonstrated in a quadrotor attitude control system in SO(3) using various datasets, including measurements collected from the AirSim simulator and the open-source NeuroBEM dataset. Furthermore, the control efficacy of the soDNN-trained system is evaluated using sequential linear model predictive control (sLMPC).