In this paper, we introduce a new and efficient way to compute exactly divergence-free velocity approximations for the Stokes equations in two space dimensions. We begin by considering a mixed method that provides an exactly divergence-free approximation of the velocity and a continuous approximation of the vorticity. We then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique. This novel formulation bypasses the difficult task of constructing an exactly divergence-free basis for velocity approximations. Moreover, the discrete system resulting from our method has fewer degrees of freedom than the original mixed method since the pressure and the tangential velocity variables are defined just on the mesh edges. Once these variables are computed, the velocity approximation satisfying the incompressibility condition exactly, as well as the continuous numerical approximation of the vorticity, can at once be obtained locally. Moreover, a discontinuous numerical approximation of the pressure within elements can also be obtained locally. We show how to compute the matrix system for our tangential velocity-pressure formulation on general meshes and present in full detail such computations for the lowest-order case of our method.