You can use the second syntax of Transpose and it should work like the code below, but it is yet to be checked whether each 4-length List as the second argument of Transpose exactly corresponds to your order of indexes or not

n = {a, b};
id4 = TensorProduct[IdentityMatrix[2], n, n];
mat = TensorProduct[n, n, n, n] - (id4 + Transpose[id4, {1, 3, 2, 4}] + 
     Transpose[id4, {1, 4, 3, 2}] + Transpose[id4, {3, 1, 4, 2}] + 
     Transpose[id4, {3, 1, 2, 4}] + Transpose[id4, {3, 4, 1, 2}])/6

You can use the second syntax of Transpose and it should work like the code below, but it is yet to be checked whether each 4-length List as the second argument of Transpose exactly corresponds to your order of indexes or not

n = {a, b};
id4 = TensorProduct[IdentityMatrix[2], n, n];
mat = TensorProduct[n, n, n, n] - (id4 + Transpose[id4, {1, 3, 2, 4}] + 
     Transpose[id4, {1, 4, 3, 2}] + Transpose[id4, {3, 1, 4, 2}] + 
     Transpose[id4, {3, 1, 2, 4}] + Transpose[id4, {3, 4, 1, 2}])/6

Update

As @WReach mentioned in the comment, two groups of indices above should indeed be corrected as below:

{3, 1, 4, 2} -> {2, 4, 1, 3}
{3, 1, 2, 4} -> {2, 3, 1, 4}

Though, the spirit of using Transpose is uninfluenced all the way.

You can use the second syntax of Transpose and it should work like the code below, but I am not sure for the each 4-length List as the second argument of Transpose

n = {a, b};
id = IdentityMatrix[2]; id4 = TensorProduct[id, n, n];
mat = TensorProduct[n, n, n, n] - (id4 + Transpose[id4, {1, 3, 2, 4}] + 
     Transpose[id4, {1, 4, 3, 2}] + Transpose[id4, {3, 1, 4, 2}] + 
     Transpose[id4, {3, 1, 2, 4}] + Transpose[id4, {3, 4, 1, 2}])/6

You can use the second syntax of Transpose and it should work like the code below, but it is yet to be checked whether each 4-length List as the second argument of Transpose exactly corresponds to your order of indexes or not

n = {a, b};
id4 = TensorProduct[IdentityMatrix[2], n, n];
mat = TensorProduct[n, n, n, n] - (id4 + Transpose[id4, {1, 3, 2, 4}] + 
     Transpose[id4, {1, 4, 3, 2}] + Transpose[id4, {3, 1, 4, 2}] + 
     Transpose[id4, {3, 1, 2, 4}] + Transpose[id4, {3, 4, 1, 2}])/6