This paper presents a cascaded control architecture, based on nonlinear dynamic inversion (NDI), for rigid body attitude control. The proposed controller works directly with the rotation matrix parameterization, that is, with elements of the Special Orthogonal Group SO(3), and avoids problems related to singularities and non-uniqueness which affect other commonly used attitude representations such as Euler angles, unit quaternions, modified Rodrigues parameters, etc. The proposed NDI-based controller is capable of imposing desired linear dynamics of any order for the outer attitude loop and the inner rate loop, and gives control designers the flexibility to choose higher-order dynamic compensators in both loops. In addition, sufficient conditions are presented in the form of linear matrix inequalities (LMIs) which ensure that the outer loop controller renders the attitude loop almost globally asymptotically stable (AGAS) and the rate loop globally asymptotically stable (GAS). Furthermore, the overall cascaded control architecture is shown to be AGAS in the case of attitude error regulation. Lastly, the proposed scheme is compared with an Euler angles-based NDI scheme from literature for a tracking problem involving agile maneuvering of a multicopter in a high-fidelity nonlinear simulation.