We present the first measurement of the Q2 dependence of the neutron spin structure function g2(n) at five kinematic points covering 0.57 (GeV/c)2 < or = Q2 < or = 1.34 (GeV/c)2 at x approximately = 0.2. Though the naive quark-parton model predicts g2 = 0, nonzero values occur in more realistic models of the nucleon which include quark-gluon correlations, finite quark masses, or orbital angular momentum. When scattering from a noninteracting quark, g2(n) can be predicted using next-to-leading order fits to world data for g1(n). Deviations from this prediction provide an opportunity to examine QCD dynamics in nucleon structure. Our results show a positive deviation from this prediction at lower Q2, indicating that contributions such as quark-gluon interactions may be important. Precision data obtained for g1(n) are consistent with next-to-leading order fits to world data.