  We study the three-body recombination of identical fermionic atoms. Using a
zero-range model for the $p$-wave interaction, we show that the rate constant
of three-body recombination into weakly bound $p$-wave dimers can be written as
$\alpha_{\rm rec} \propto v^{5/2}R^{1/2} k_T^4 (1+ C k_T^2 l_{\rm d}^2)$ for
large and positive scattering volume $v$. Here $R$ is the $p$-wave effective
range, $k_T^2$ gives the average thermal kinetic energy of the colliding atoms,
and $l_{\rm d}$ is the size of the $p$-wave dimer. The leading term is
different from the usually stated $v^{8/3}$-scaling law, but is consistent with
an earlier two-channel calculation. For the subleading term, we compute the
constant $C$ by solving the relevant three-body problem perturbatively when the
parameter $\gamma\equiv R/v^{1/3}$ is small. The additional $C k_T^2 l_{\rm
d}^2$ term provides important corrections for the temperature and interaction
dependence of $\alpha_{\rm rec}$, especially close to resonance when $k_T
l_{\rm d}$ is relatively large.
