In network coding, a flag code is a set of sequences of nested subspaces of F_q^n, being F_q the finite field of size q. Flag codes defined as orbits of a cyclic subgroup of the general linear group acting on flags of F_q^n are called cyclic orbit flag codes. In this work, we completely characterize those cyclic orbit flag codes attaining both the best possible size and distance, that is, optimal full-length cyclic orbit flag codes in terms of the generating flag. As a consequence, we can provide the distance distribution of this family of codes.