The idea of the Gauss map is unified with the concept of branes as hypersurfaces embedded into D-dimensional Minkowski space. Themap introduces new generalized coordinates of branes alternative to their world vectors x and identified with the gauge and other massless fields. In these coordinates the Dirac p-branes realize extremals of the Euler-Lagrange equations of motion of a (p + 1)dimensional SO(D - p - 1) gauge-invariant action in a gravitational background.