We use the LDA+U method to study the possibility of exciton condensation in perovskites of transition metals with d^6 electronic configuration such as LaCoO_3. For realistic interaction parameters we find several distinct solutions exhibiting spin-triplet exciton condensate, which gives rise to a local spin density distribution while the ordered moments are vanishingly small. Rhombohedral distortion from the ideal cubic structure suppresses the ordered state, contrary to the spin-orbit coupling

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... which enhances the excitonic condensation energy. We explain the trends observed in the numerical simulations with the help of a simplified strong-coupling model. Our results indicate that LaCoO_3 is close to the excitonic instability and suggest ways to make it order.