The observation of metallic behavior at the interface between insulating oxides has triggered worldwide efforts to shed light on the physics of these systems and clarify some still open issues, among which the dimensional character of the conducting system. In order to address this issue, we measure electrical transport (Seebeck effect, Hall effect and conductivity) in LaAlO_{3}/SrTiO_{3} interfaces and, for comparison, in a doped SrTiO_{3} bulk single crystal. In these experiments, the carrier concentration is tuned, using the field effect in a back gate geometry. The combined analysis of all experimental data at 77 K indicates that the thickness of the conducting layer is ~7 nm and that the Seebeck effect data are well described by a two-dimensional (2D) density of states. We find that the back gate voltage is effective in varying not only the charge density, but also the thickness of the conducting layer, which is found to change by a factor of ~2, using an electric field between -4 and +4MV/m at 77K. No enhancement of the Seebeck effect due to the electronic confinement and no evidence for two-dimensional quantization steps are observed at the interfaces. ; Comment: 15 pages, 5 figures