In this paper we investigate the equations for light-like extremal surfaces in Minkowski space R^{1+(1+n)}. We show that the light-like assumption is compatible with the Cauchy problem and give a necessary and sufficient condition on the global existence of classical solutions of the Cauchy problem. Based on this, we obtain entire light-like extremal surfaces by solving the Cauchy problem explicitly when such necessary and sufficient condition holds. Finally, some discussions and related remarks are given.