We plan to develop an advanced Transition Radiation Detector (TRD) for hadron identification in the TeV momentum range, based on the simultaneous measurement of the energies and of the emission angles of the Transition Radiation (TR) X-rays with respect to the radiating particles. To study the feasibility of this project, we have carried out a beam test campaign at the CERN SPS facility with 20 GeV/c electrons and muons up to 300 GeV/c. To detect the TR X-rays and the radiating particles, we used a 300 µm thick double-sided silicon strip detector, with a strip pitch of 50 µm. A 2 m long helium pipe was placed between the radiators and the detector, in order to ensure adequate separation between the TR X-rays and the radiating particle on the detector plane and to limit the X-ray absorption before the detector. We measured the double-differential (in energy and angle) spectra of the TR emitted by several radiators. The results are in good agreement with the predictions obtained from the TR theory.