In this paper, we present a novel distributed method to stabilize a set of agents moving in a two dimensional environment to a desired rigid formation. In our approach, each agent computes its control input using the relative positions of a set of formation neighbors but, contrary to most existing works, this information is expressed in the agent's own independent local coordinate frame, without requiring any common reference. The controller is based on the minimization of a Lyapunov function that includes locally computed rotation matrices, which are required due to the absence of a common orientation. Our contribution is that the proposed distributed coordinate-free method achieves global stabilization to a rigid formation with the agents using only partial information of the team, does not require any leader units, and is applicable to both single-integrator or unicycle agents. To guarantee global stability, we require that the network induced by the agent interactions belongs to a certain class of undirected rigid graphs in two dimensions, which we explicitly characterize. The performance of the proposed method is illustrated with numerical simulations.