We use the Gemini NIFS instrument to map the H$_2 2.1218mu$m and Br$gamma$ flux distributions in the inner 0.04-2 kpc of a sample of 36 nearby active galaxies ($0.001lesssim zlesssim0.056$) at spatial resolutions from 4 to 250 pc. We find extended emission in 34 galaxies. In $sim$55% of them, the emission in both lines is most extended along the galaxy major axis, while in the other 45% the extent follows a distinct orientation. The emission of H$_2$ is less concentrated than that of Br$gamma$, presenting a radius that contains half of the flux 60% greater, on average. The H$_2$ emission is driven by thermal processes - X-ray heating and shocks - at most locations for all galaxies, where $0.4<H_2/Brgamma<6$. For regions where H$_2$/Br$gamma>6$ (seen in 40% of the galaxies), shocks are the main H$_2$ excitation mechanism, while in regions with H$_2$/Br$gamma<0.4$ (25% of the sample) the H$_2$ emission is produced by fluorescence. The only difference we found between type 1 and type 2 AGN was in the nuclear emission-line equivalent widths, that are smaller in type 1 than in type 2 due to a larger contribution to the continuum from the hot dusty torus in the former. The gas masses in the inner 125 pc radius are in the range $10^1-10^4$ M$_odot$ for the hot H$_2$ and $10^3-10^6$ M$_odot$ for the ionised gas and would be enough to power the AGN in our sample for $10^5-10^8$ yr at their current accretion rates.
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