The Leibniz formula for determinants starts with multi-linearity and the alternating property and builds from there. I asked a question about why we should start with multi-linearity: What's so special about multi-linearity?. And the response (which had an excellent alternate motivation for the determinant which I wasn't aware of) seemed to be that there is indeed no good reason.
The thing that keeps bothering me is that we are talking about linear maps here. And the determinant is really a property of the linear map. And given the definitions of linearity in the context of linear maps and the multi-linearity of determinants look exactly the same, I would assume a statement like this exists - "because the determinant is a property of linear maps, it clearly must satisfy the property of multi-linearity because _____". Is there no way to fill in the blank?