Specific rearrangement of elements in list

Since it seems to suit the OP's needs:

G = Array[g, {4, 4}];
Join @@ Table[-Differences[Diagonal[G, k]], {k, {0, 1, -1}}]
   {g[1, 1] - g[2, 2], g[2, 2] - g[3, 3], g[3, 3] - g[4, 4], g[1, 2] - g[2, 3],
    g[2, 3] - g[3, 4], g[2, 1] - g[3, 2], g[3, 2] - g[4, 3]}

The key here is the use of Diagonal[] to extract the diagonals of a matrix (note the second argument for picking which diagonal to extract), and the use of Differences[] to generate the needed entries; however, the sign is opposite from what the OP needed, so we perform a negation afterwards. Join[] merely strings together the three diagonals thus extracted into a single list.

k = Table[g[i, j], {i, 4}, {j, 4}]; 
Flatten[Diagonal[k[[;; -2, ;; -2]] - k[[2 ;;, 2 ;;]], #] & /@ {0, 1, -1}]