We report molecular dynamics simulations of a Gay-Berne nematic liquid crystal at constant temperature and density/pressure using the generalization of an algorithm recently proposed by Toxvaerd [Phys. Rev. E 47, 343 (1993)]. On the basis of these simulations the absolute values of the Oseen-Zocher-Frank elastic constants K-11, K-22, and K-33 as well as the surface constants K-13 and K-24 have been calculated ab initio within the framework of the direct correlation function approach of Lipkin et al. [J. Chem. Phys. 82, 472 (1985)]. The angular coefficients of the direct pair correlation function, which enter the final equations, have been determined from the computer simulation data for the pair correlation function of the nematic by combining the Ornstein-Zernike relation and the Wiener-Hopf factorization scheme. The unoriented nematic approximation has been assumed when constructing the reference state of Lipkin et al. By an extensive study of the model over a wide range of temperatures, densities and pressures, very detailed information is provided on the elastic behavior of the Gay-Berne nematic. Interestingly, it is found that the results for the surface elastic constants are qualitatively different from those obtained with the help of analytical approximations for the isotropic direct pair correlation function. For example, the values of the surface elastic constants are partly negative and an order of magnitude smaller than the bulk elasticity. The negative values of the surface constant K,, indicate on the possibility of surface instabilities of the director pattern in a thin, free standing or weakly anchored Gay-Berne nematic liquid crystal.