I am interested in $n \times n$ matrices over some field $K$ all whose rows and all whose columns sum to zero.

First question: do these matrices have a name?

Pending an answer I will call these "null-matrices".

Second (main) question:

Given $n$, are there subsets $J \subset \{1, \ldots n\} \times \{1, \ldots, n\}$ of indices such that
$$\sum_{(i, j) \in J} a_{i,j} = 0$$
for *every* null-matrix $(a_{ij})_{i, j = 1 \ldots n}$?

More visually: can you take a red pencil and put red circles around some entries in a an (still empty) matrix so that in whichever way someone fills up the matrix with elements of $K$ to obtain a null-matrix, the sum of the red-circled entries will always add up to zero?

Obviously the answer is yes, just take $J$ to be a disjoint union of rows or a disjoint union of columns. So my question is: are there examples of set $J$ that are not of this form?

In general (i.e. without specifying the field over which we consider the matrix) I expect the answer to be no (but please prove me wrong) - however for some special combinations of $n$ and char($K$) the answer might be yes. In particular, for $n = 2$, char($K$) $=2$, the diagonal (i.e. $J = \{(1, 1), (2,2)\}$) is an example of the type of set I'm looking for.

Are there more examples like this?