Magnetic Cooper-pairing mechanisms have been proposed for heavy-fermion and cuprate superconductors; however, strong electron correlations(1) and complications arising from a pseudogap(2-4) or competing phases(5) have precluded commonly accepted theories. In the iron arsenides, the proximity of superconductivity and antiferromagnetism in the phase diagram(6,7), the apparently weak electron-phonon coupling(8) and the 'resonance peak' in the superconducting spin-excitation spectrum(9-11) have also fostered the hypothesis of magnetically mediated Cooper pairing. However, as most theories of superconductivity are based on a pairing boson of sufficient spectral weight in the normal state, detailed knowledge of the spin-excitation spectrum above the superconducting transition temperature T-c is required to assess the viability of this hypothesis(12,13). Using inelastic neutron scattering we have studied the spin excitations in optimally doped BaFe1.85Co0.15As2 (T-c = 25 K) over a wide range of temperatures and energies. We present the results in absolute units and find that the normal-state spectrum carries a weight comparable to that in underdoped cuprates(14,15). In contrast to cuprates, however, the spectrum agrees well with predictions of the theory of nearly antiferromagnetic metals(16), without the aforementioned complications. We also show that the temperature evolution of the resonance energy monotonically follows the closing of the superconducting energy gap Delta, as expected from conventional Fermi-liquid approaches(17,18). Our observations point to a surprisingly simple theoretical description of the spin dynamics in the iron arsenides and provide a solid foundation for models of magnetically mediated superconductivity.