How to write this tensor?

The simplest way would be to use Table, e.g. something like

tensorTable[n_] := With[{dim = Length[n]},
  Table[
   (KroneckerDelta[i, k] - n[[i]] n[[k]]) (KroneckerDelta[j, l] - 
       n[[j]] n[[l]]) -
    1/2 (KroneckerDelta[i, j] - n[[i]] n[[j]]) (KroneckerDelta[k, l] -
        n[[k]] n[[l]]),
   {i, dim}, {j, dim}, {k, dim}, {l, dim}]]

For the vector given by OP:

exampleTensor = Simplify[tensorTable[{Cos[ϕ] Sin[θ],Sin[ϕ] Sin[θ], Cos[θ]}]];

I suppose you could also write this in more convenient(?) mathematical notation using SparseArray, by noticing the two terms are identical up to a permutation of $jleftrightarrow k$

commonTerm[n_] := With[{dim = Length[n]},
  Plus @@ Map[SparseArray[#, {dim, dim, dim, dim}] &,
    {
     {i_, j_, k_, l_} :> n[[i]] n[[j]] n[[k]] n[[l]],
     {i_, j_, i_, l_} :> -n[[j]] n[[l]],
     {i_, j_, k_, j_} :> -n[[i]] n[[k]],
     {i_, j_, i_, j_} :> 1
     }
    ]
  ]

This will also store the tensor as a SparseArray which may be beneficial for further analysis.

exampleTensor2 = 
  Simplify[commonTerm[{Cos[ϕ] Sin[θ],Sin[ϕ] Sin[θ], Cos[θ]}] - 
    1/2 Transpose[commonTerm[{Cos[ϕ] Sin[θ], Sin[ϕ] Sin[θ], Cos[θ]}], 
      Cycles[{{2, 3}}]]];

exampleTensor == exampleTensor2
(*True*)