  We lay the foundation for determining the microscopic spin interactions in
two-dimensional (2D) ferromagnets by combining angle-dependent ferromagnetic
resonance (FMR) experiments on high quality CrI$_3$ single crystals with
theoretical modeling based on symmetries. We discover that the Kitaev
interaction is the strongest in this material with $K \sim -5.2$ meV, 25 times
larger than the Heisenberg exchange $J \sim -0.2$ meV, and responsible for
opening the $\sim$5 meV gap at the Dirac points in the spin-wave dispersion.
Furthermore, we find that the symmetric off-diagonal anisotropy $\Gamma \sim
-67.5$ $\mu$eV, though small, is crucial for opening a $\sim$0.3 meV gap in the
magnon spectrum at the zone center and stabilizing ferromagnetism in the 2D
limit. The high resolution of the FMR data further reveals a $\mu$eV-scale
quadrupolar contribution to the $S=3/2$ magnetism. Our identification of the
underlying exchange anisotropies opens paths toward 2D ferromagnets with higher
$T_\text{C}$ as well as magnetically frustrated quantum spin liquids based on
Kitaev physics.
