I've addressed this problem with MILP. For $n\leq 5$, the maximum happen to be just $n$, but for $n=6$ it's only $\frac{16}3$. Here are the constructed solutions:

    Max: 3
    [ 0 -1  1]
    [-1  1  0]
    [ 1  0 -1]

    Max: 4
    [ 1/4 -1/4 -3/4  3/4]
    [ 3/4  1/4 -1/4 -3/4]
    [-3/4  3/4  1/4 -1/4]
    [-1/4 -3/4  3/4  1/4]

    Max: 5
    [ 1/2  1/2 -1/2 -1/2    0]
    [   0 -1/2 -1/2  1/2  1/2]
    [-1/2    0  1/2 -1/2  1/2]
    [-1/2  1/2    0  1/2 -1/2]
    [ 1/2 -1/2  1/2    0 -1/2]

    Max: 16/3
    [ -3/8  -1/3   3/8 11/24 -7/24   1/6]
    [11/24  -1/3 -7/24  -3/8   3/8   1/6]
    [  3/8   1/6 11/24 -7/24  -3/8  -1/3]
    [-7/24   1/6  -3/8   3/8 11/24  -1/3]
    [ -1/3   1/6   1/6  -1/3   1/6   1/6]
    [  1/6   1/6  -1/3   1/6  -1/3   1/6]