I think Dan Piponi's point is that $D$ is a finite discrete approximation to a continuous operator. As you approach the continuum limit the harder it gets to tell the difference between $D^T$ and $-D$ "from a distance" so to speak. Both have a diagonal line of -1s adjacent to a diagonal line of 1s just below it. The difference is which a small shift as to which me is the actual diagonal.
I think his argument can be simplified if $D$ had the -1s on the superdiagonal instead of the diagonal. It would still be a valid approximation of the differential, and it really would be antisymmetric.