Context. Knowledge of the mass function in open clusters constitutes one way to critically examine the formation mechanisms proposed to explain the existence of low-mass stars and brown dwarfs. Aims: The aim of the project is to determine as accurately as possible the shape of the mass function across the stellar/substellar boundary in the young (5 Myr) and nearby (d = 145 pc) Upper Sco association. Methods: We have obtained multi-fibre intermediate-resolution (R ~ 1100) optical (~5750-8800 Å) spectroscopy of 94 photometric and proper motion selected low-mass star and brown dwarf candidates in Upper Sco with the AAOmega spectrograph on the Anglo-Australian Telescope. Results: We have estimated the spectral types and measured the equivalent widths of youth (Hα) and gravity (Na I and K I) diagnostic features to confirm the spectroscopic membership of about 95% of the photometric and proper motion candidates extracted from 6.5 square degrees surveyed in Upper Sco by the UKIRT Infrared Deep Sky Survey (UKIDSS) Galactic Clusters Survey (GCS). We also detect lithium in the spectra with the highest signal-to-noise, consolidating our conclusions about their youth. Furthermore, we derive an estimate of the efficiency of the photometric and proper motion selections used in our earlier studies using spectroscopic data obtained for a large number of stars falling into the instrument's field-of-view. We have estimated the effective temperatures and masses for each new spectroscopic member using the latest evolutionary models available for low-mass stars and brown dwarfs. Combining the current optical spectroscopy presented here with near-infrared spectroscopy obtained for the faintest photometric candidates, we confirm the shape and slope of our earlier photometric mass function. The luminosity function drawn from the spectroscopic sample of 113 USco members peaks at around M6 and is flat at later spectral type. We may detect the presence of the M7/M8 gap in the luminosity function as a result of the dust properties in substellar atmospheres. The mass function may peak at 0.2 M&sun; and is quite flat in the substellar regime. We observe a possible excess of cool low-mass brown dwarfs compared to IC 348 and the extrapolation of the field mass functions, supporting the original hypothesis that Upper Sco may possess an excess of brown dwarfs compared to other young regions. Conclusions: This result shows that the selection of photometric candidates based on five band photometry available from the UKIDSS GCS and complemented partially by proper motions can lead to a good representation of the spectroscopic mass function. Based on observations obtained with the AAOmega spectrograph at the Anglo-Australian Observatory.Appendices are only available in electronic form at http://www.aanda.orgFull Table B.1 and optical spectra are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/527/A24