We report on trapping of fermionic ${}^{40}\mathrm{K}$ atoms in a red-detuned standing-wave optical trap, loaded from a magneto-optical trap. Typically, unpolarized samples of ${10}^{6}$ atoms are loaded at a density of ${10}^{12}{\mathrm{cm}}^{\ensuremath{-}3}$ and a temperature of $65\ensuremath{\mu}\mathrm{K},$ and trapped for more than 1 s. The optical trap appears to be the proper environment for performing collisional measurements on the cold atomic sample. In particular, we measure the spherically averaged elastic collision rate by detecting the rethermalization following an intentional parametric heating of the atoms. We also measure spherically averaged inelastic two-body collision rates for atoms in the ground hyperfine states, through detection of trap losses.