In this paper, we study surfaces in $mathb{E}^3$ whose Gauss map G satisfies the equation ${Box}G=f(G+C)$ for a smooth function $f$ and a constant vector C, where ${Box}$ stands for the Cheng-Yau operator. We focus on surfaces with constant Gaussian curvature, constant mean curvature and constant principal curvature with such a property. We obtain some classification and characterization theorems for these kinds of surfaces. Finally, we give a characterization of surfaces whose Gauss map G satisfies the equation ${Box}G={lambda}(G+C)$ for a constant ${lambda}$ and a constant vector C.