The localization properties of a two-dimensional electron system in the presence of a random magnetic field are numerically studied by calculating directly eigenstates of large systems. Analyzing eigenstates themselves gives precise information on the localization of wave functions compared with that obtained from transport properties. The finite-size scaling analysis for quasi-one-dimensional wave functions clearly shows that the single-parameter scaling hypothesis is valid for this system and all states are localized. Furthermore, an analysis based on the multifractality of a localized wave function also supports these results.