  Using single crystal inelastic neutron scattering with and without
application of an external magnetic field and powder neutron diffraction, we
have characterized magnetic interactions in Ba$_3$Cr$_2$O$_8$. Even without
field, we found that there exist three singlet-to-triplet excitation modes in
$(h,h,l)$ scattering plane. Our complete analysis shows that the three modes
are due to spatially anisotropic interdimer interactions that are induced by
local distortions of the tetrahedron of oxygens surrounding the Jahn-Teller
active Cr$^{5+} (3d^1)$. The strong intradimer coupling of $J_0 = 2.38(2)$ meV
and weak interdimer interactions ($|J_{\rm inter}| \leq 0.52(2)$ meV) makes
Ba$_3$Cr$_2$O$_8$ a good model system for weakly-coupled $s = 1/2$ quantum spin
dimers.
