Solution:

Answer: no. Fix an integer $k$ from $1$ to $4021$. Let us say that an index $i$ "hits" a cell containing the number $k$ if the cell is in either the $i$th row or the $i$th column. The conditions stipulate that each index hits exactly one instance of $k$, so the total number of hits is $4021$. On the other hand, every cell not lying on the main diagonal is hit by exactly two indices (its row number and its column number), while those on the diagonal are hit only once. In particular, to create an odd total number of hits, $k$ must appear on the diagonal. This is a contradiction since there are $4021$ permissible values of $k$ and only $2011$ spots on the diagonal.