Lane changing is one of the most common maneuvers on motorways. Although macroscopic traffic models are well known for their suitability to describe fast moving crowded traffic, most of these models are generally developed in a one-dimensional framework; henceforth lane-changing behavior is either not explicitly modeled or is explicitly forbidden. In this paper, we propose a macroscopic model which accounts for lane-changing behavior on motorways, based on a two-dimensional extension of the Aw and Rascle [SIAM J. Appl. Math., 60 (2000), pp. 916--938] and Zhang [Transport. Res. B-Meth., 36 (2002), pp. 275--290] macroscopic model for traffic flow. Under conditions when lane-changing maneuvers are no longer possible, the model “relaxes” to the one-dimensional Aw--Rascle--Zhang model. Following the same approach as in [SIAM J. Appl. Math., 63 (2002), pp. 259--278], we derive the two-dimensional macroscopic model through scaling of time discretization of a microscopic follow-the-leader model with driving direction. We provide a detailed analysis of the space-time discretization of the proposed macroscopic model as well as an approximation of the solution to the associated Riemann problem. Furthermore, we illustrate some promising features of the proposed model through some numerical experiments.