We investigate the transition of the turbulence from large to kinetic scales using Cluster observations. Simultaneous spectra of magnetic and electric fields in the Earth's magnetosheath from magnetohydrodynamic (MHD) to electron scales are presented for the first time. While the two spectra have approximatively similar behaviour in the fluid-MHD regime, they show different trends in the kinetic range. As the magnetic field spectrum steepens at ion scales, the electric field spectrum is characterized by a shallower power law continuing down to electron scales. Such an evolution is consistent with theoretical expectations, assuming that the turbulence is dominated by highly oblique k-vectors and that between ion and electron scales the electric field is governed by the non-ideal terms in the generalized Ohm's law. This leads to an expected linear increase of the electric-to-magnetic ratio of fluctuations, consistent with observations presented here. The influence of local whistler wave activity on electron-scale spectra is also discussed.