The primal LP for $n=3$ looks like this:
$$
\max 1^Tx\\
\text{s.t. }
\left[\matrix{
1 & 1 & 0\\
1 & 0 & 1\\
0 & 1 & 1\\
0 & 0 & 1\\
}\right]
\left[\matrix{
x_0\\x_1\\x_2
}\right]
\leq
\left[\matrix{
d_{01}\\
d_{02}\\
d_{11}\\
d_{22}\\
}\right]\\
x\geq 0
$$
Note, that the definition is not well defined, as $j\in[n,n)$ makes not much sense, so I just removed the last constraint from the set of possible combinations of $i$ and $j$. If your definition is meant to be understood differently, just remove the corresponding constraints.
Applying the standard dualization rules results in this dual LP:
$$
\min \left[\matrix{d_{01} & d_{02} & d_{11} & d_{22}}\right]^Ty\\
\text{s.t. }
\left[\matrix{
1 & 1 & 0 & 0\\
1 & 0 & 1 & 0\\
0 & 1 & 1 & 1\\
}\right]
\left[\matrix{
y_0\\y_1\\y_2\\y_3
}\right]
\geq
\left[\matrix{
1\\
1\\
1\\
}\right]\\
y\geq 0
$$
The constraint matrix of the dual is simply $A^T$ so the general formulation is then:
$$
\min d^Ty\\
\text{s.t. }
y_j + y_i \geq 1, \; j\in[0,m-1], \; i\in[j+1,m-1]\\
y \geq 0
$$
The size of $m$ is not completely clear as I am not sure about your initial formulation.