We present new experimental results of the $^3$He spin structure function
$g_2$ in the resonance region at $Q^2$ values between 1.2 and 3.0 (GeV/c)$^2$.
Spin dependent moments of the neutron were then extracted. Our main result, the
resonance contribution to the neutron $d_2$ matrix element, was found to be
small at $
$=2.4 (GeV/c)$^2$ and in agreement with the Lattice QCD
calculation. The Burkhardt-Cottingham sum rule for $^3$He and the neutron was
tested with the measured data and using the Wandzura-Wilczek relation for the
low $x$ unmeasured region. A small deviation was observed at $Q^2$ values
between 0.5 and 1.2 (GeV/c)$^2$ for the neutron.