Hi Dick, I don't know if there is one or another way to give a formal meaning to this formula, but I understand it this way:

We have the identity $\det[X\ Y\ Z] = \langle X, Y \times Z\rangle$, where the brackets denotes the scalar product. So, $Y \times Z = \sum_{i=1}^3 \langle e_i, Y \times Z \rangle e_i = \sum_{i=1}^3 \det[e_i \ Y\ Z]e_i$, where $e_i$ are the vectors of the canonical basis, which is your formula, I think. 

-- Note: It seems to be the same remark as Deane Yang's