  A second-order formalism for the theory of 3D spin-3 gravity is considered.
Such a formalism is obtained by solving the torsion-free condition for the spin
connection \omega^a_{\mu}, and substituting the result into the action
integral. In the first-order formalism of the spin-3 gravity defined in terms
of SL(3,R) X SL(3,R) Chern-Simons (CS) theory, however, the generalized
torsion-free condition cannot be easily solved for the spin connection, because
the vielbein e^a_{\mu} itself is not invertible. To circumvent this problem,
extra vielbein-like fields e^a_{\mu\nu} are introduced as a functional of
e^a_{\mu}. New set of affine-like connections \Gamma_{\mu M}^N are defined in
terms of the metric-like fields, and a generalization of the Riemann curvature
tensor is also presented. In terms of this generalized Riemann tensor the
action integral in the second-order formalism is expressed. The transformation
rules of the metric and the spin-3 gauge field under the generalized
diffeomorphims are obtained explicitly. As in Einstein gravity, the new
affine-like connections are related to the spin connection by a certain gauge
transformation, and a gravitational CS term expressed in terms of the new
connections is also presented.
