The development of a nanosecond surface dielectric barrier discharge in air at pressures 1-6 bar is studied. At atmospheric pressure, the discharge develops as a set of streamers starting synchronously from the high-voltage electrode and propagating along the dielectric layer. Streamers cover the dielectric surface creating a 'quasi-uniform' plasma layer. At high pressures and high voltage amplitudes on the cathode, filamentation of the discharge is observed a few nanoseconds after the discharge starts. Parameters of the observed 'streamers-to-filaments' transition are measured; physics of transition is discussed on the basis of theoretical estimates and numerical modeling. Ionization-heating instability on the boundary of the cathode layer is suggested as a mechanism of filamentation.