An exponential parametrization of the one-electron atomic-orbital density matrix of Hartree-Fock and Kohn-Sham theories is proposed. The parametrization is non-redundant and satisfies the usual symmetry, trace and idempotency conditions exactly, eliminating the need to enforce these conditions explicitly during the optimization of the energy or the calculation of molecular properties. The exponential parametrization makes the construction of the (transformed) density matrix simple, requiring only elementary matrix operations such as additions and multiplications; it is therefore ideally suited to the study of large molecular systems at the Hartree-Fock and Kohn-Sham levels of theory, at a cost that scales linearly with the size of the system.