First, note that your edited question still has $i=2n$ when you mean $i=n$. As was pointed out by others, $n$ is the rank here and $2n$ the dimension of the first fundamental representation (the natural one for the Lie algebra in the even orthogonal case).
One useful source (if you can locate it), based on lectures given by J. Tits to mathematical physicists in Bonn many years ago, is his Springer Lecture Notes No. 40 (1967) with a German title Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen [Tables for the simple Lie groups and their representations]. But the tables toward the end of this short volume involve little German (and Tits himself is Belgian/French). He provides explicit data including dimensions of all the fundamental representations, describing also the results for the various real forms.
In particular, the two end vertices of the Dynkin diagram usually labelled $n-1$ and $n$, correspond to the half-spin representations of the Lie algebra. Each has dimension $2^{n-1}$. They are not so easily constructed in terms of exterior powers of the natural representation as the earlier fundamental representations, however, with a digression into Clifford algebras usually required.