An analytical solution is obtained for the wind-driven steady flow developing under the action of the Coriolis acceleration in a closed basin of elongated shape. Different from the traditional Ekman approach, which determines the velocity distribution along a water column given the free surface shear stress and pressure gradient, here the flow field is solved in the whole cross-section considering the lateral transfer of momentum due to the horizontal eddy viscosity. The solution is derived exploiting a perturbation method, whereby the inverse of the Ekman number is assumed small, and imposing a wind aligned with the main axis of the lake. In the central part of the lake a secondary circulation develops producing downwelling along the right hand side (in the northern hemisphere) and upwelling along the opposite side, whose intensity is modulated by the turbulence anisotropy. The modification of the primary flow is considered as well. The solution, which is also compared with numerical results, is obtained for simplified conditions, but the extension to more general cases is discussed.