A new formulation of the virial expansion for a classical gas is derived without the restriction to pairwise-additive interaction potentials. Explicit expressions up to the eighth virial coefficient, suitable for numerical evaluation, are given in the form of integrals over sums of cluster diagrams. Compared with previous approaches, the number of cluster diagrams increases more moderately with increasing order of the virial coefficient. Thus, the new formulation should be particularly useful for the computation of high-order virial coefficients.