We have performed multi-frequency electron spin resonance (ESR) measurements on EuFe2As2, which is one of the parent compounds of Fe-based superconductors. At T SDW=190 K, this compound shows a spin-density-wave (SDW) transition. At temperatures above T SDW, the relaxation of Eu2+ spins takes place via the coupling of conduction electrons, resulting in a Korringa-type increase in the linewidth. At temperatures below T SDW, however, this compound shows an anisotropic temperature-independent linewidth except at temperatures near the long-range ordering temperature (19 K) of Eu2+ spins, giving evidence for a gap opening at T SDW. This change of physical properties indicates a change in the exchange interaction between localized Eu2+ spins and conduction electrons in the FeAs layers. We evaluate the exchange constant, which was not reported in previous X-band ESR studies, and compare it with the value for another Eu metallic compound.