The combination of high optical transparency and low electrical resistivity has made transparent conductive oxides (TCOs) a key technology in many optoelectronic applications. Furthermore, the study of TCOs yields insight into many fundamental parameters of semiconductors. For example, the high charge carrier concentration results in an apparent shift in the band gap, the so-called Burstein-Moss shift, in addition to plasmonic resonances in the near infrared regime. While both effects are related to the carrier concentration and band structure, their lateral distribution and interaction with boundary conditions such as interfaces and surfaces are difficult to assess, and a direct observation of the local distribution has remained elusive. Here we employ electron energy-loss spectroscopy in scanning transmission electron microscopy (STEM-EELS) for direct observation of spatially resolved plasmonic resonances and Burstein-Moss shift in gallium doped zinc oxide (GZO), one of the most widely used TCOs. A 25 nm thick GZO film with a carrier concentration of $7\ifmmode\times\else\texttimes\fi{}{10}^{20}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$ has been grown epitaxially on a nominally undoped ZnO film. The GZO film shows a renormalized Burstein-Moss shift of $\ensuremath{\sim}0.5$ eV, in accordance with that expected from Hall effect measurements. The plasma resonance of the conduction band electrons is located at 0.82 eV in the bulk GZO, and with a substantial increase in intensity towards the sample surface, where a surface plasmon energy of 0.54 eV is observed. Hence we have directly measured differences in optical properties between the two films, the local variation across the GZO film has been studied, and we are also directly observing the difference between bulk and surface properties of GZO.