Based on correspondence rules between quantum-mechanical operators and classical functions in phase space we construct exchange-energy densities in position space. Whereas these are not unique but depend on the chosen correspondence rule, the exchange potential is unique. We calculate this exchange-energy density for 15 closed-shell atoms, and compare it with kinetic-and Coulomb-energy densities. It is found that it has a dominating local-density character, but electron-shell effects are recognizable. The approximate exchange-energy functionals that have been proposed so far are found to account only poorly for the observed behaviors. Instead we use our results in proposing an alternative functional that depends on both first-and second-order derivatives of the electron density.