Will this do?
Table[ If[j<k, o[j].o[k] - o[k].o[j]], {j, 1, 3}, {k, 1, 3}] /. Null -> Sequence[] // Flatten
which gives you
{o[1].o[2] - o[2].o[1], o[1].o[3] - o[3].o[1], o[2].o[3] - o[3].o[2]}
Or a slightly simpler version
Table[If[j<k, o[j].o[k] - o[k].o[j], {}], {j, 1, 3}, {k, 1, 3}] // Flatten