Macroscopic models of inertial flows in porous media have many practical applications where direct numerical simulations are not feasible. The Forchheimer equation describes macroscopic momentum transport accounting for inertial effects at the pore scale through a non-linear correction tensor $\mathbf{F}_β$ . Using up-scaling approaches such as the volume averaging method, $\mathbf{F}_β$ can be determined through the resolution of a closure problem. However this later is non-linear and still depends on the intensity and orientation of the macroscopic flow, complicating its resolution. Moreover, while the influence of the pore Reynolds number on the Forchheimer correction has been studied in details, its dependency on the flow orientation has drawn much less attention. In this work, a system of linearized closure problem is proposed and solved, allowing to determine $\mathbf{F}_β$ without the need to solve the full closure problems for each orientation of the macroscopic pressure gradient. The validity of this approach is assessed for various rectangular unit cells against numerical solutions of the corresponding non-linear problem, showing excellent agreement for pore Reynolds number up to unity. Then macroscopic simulations are performed to evaluate the importance of varying flow orientation on the macroscopic inertial flow. Numerical results of the general non-linear macroscopic model obtained by the volume averaging method highlight the necessity to account for extra-diagonal terms as well as macroscopic gradient orientation in the determination of the Forchheimer tensor.