We present ground-based and Swift photometric and spectroscopic observations of the tidal disruption event (TDE) ASASSN-15oi, discovered at the centre of 2MASX J20390918-3045201 (d~=216Mpc) by the All-Sky Automated Survey for SuperNovae. The source peaked at a bolometric luminosity of L~=1.3x10^44^erg/s and radiated a total energy of E~=6.6x10^50^ erg over the first ~3.5 months of observations. The early optical/UV emission of the source can be fit by a blackbody with temperature increasing from T~=2x10^4^K to T~=4x10^4^K while the luminosity declines from L~=1.3x10^44^erg/s to L~=2.3x10^43^erg/s, requiring the photosphere to be shrinking rapidly. The optical/UV luminosity decline during this period is most consistent with an exponential decline, L{prop.to}e^-(t-t_0)/tau_^, with {tau}~=46.5d for t_0_~=57241.6 (MJD), while a power-law decline of L{prop.to}(t-t_0_)^-{alpha}^ with t_0_~=57212.3 and {alpha}=1.62 provides a moderately worse fit. ASASSN-15oi also exhibits roughly constant soft X-ray emission that is significantly weaker than the optical/UV emission. Spectra of the source show broad helium emission lines and strong blue continuum emission in early epochs, although these features fade rapidly and are not present ~3 months after discovery. The early spectroscopic features and colour evolution of ASASSN-15oi are consistent with a TDE, but the rapid spectral evolution is unique among optically selected TDEs.