How to generate a table with i != j

Playing with Nothing:

1) In-process:

Table[If[i == j, Nothing, {i, j}], {i, 1, 4}, {j, 1, 4}]

2) Post-process:

Replace[Table[{i, j}, {i, 1, 4}, {j, 1, 4}], {i_, i_} -> Nothing, {2}]

Delete:

a = Array[List, {4, 4}];

Delete[a, Array[{#, #} &, 4]]

{{{1, 2}, {1, 3}, {1, 4}},
 {{2, 1}, {2, 3}, {2, 4}},
 {{3, 1}, {3, 2}, {3, 4}},
 {{4, 1}, {4, 2}, {4, 3}}}

Or, inspired by kglr's answer, with Pick and IdentityMatrix:

Pick[a, IdentityMatrix[4], 0]

{{{1, 2}, {1, 3}, {1, 4}},
 {{2, 1}, {2, 3}, {2, 4}},
 {{3, 1}, {3, 2}, {3, 4}},
 {{4, 1}, {4, 2}, {4, 3}}}

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Benchmark

With many methods provided I think it is time for a benchmark. I do not have Nothing in version 10.1 so I shall use my old standby "vanishing function" from How to avoid returning a Null if there is no "else" condition in an If contruct in its place.

n = 1000;

(* for post processing methods this timing must be included in the total *)
a = Array[List, {n, n}]; // RepeatedTiming // First

DeleteCases[a, {a_, a_}, 2] // RepeatedTiming // First

Table[If[i == j, ## &[], {i, j}], {i, n}, {j, n}] // RepeatedTiming // First

Partition[#, n - 1] &@Permutations[Range@n, {2}] // RepeatedTiming // First

Select[#, DuplicateFreeQ] & /@ Outer[List, Range@n, Range@n] // 
  RepeatedTiming // First

Delete[a, Array[{#, #} &, n]] // RepeatedTiming // First

Partition[SparseArray[ConstantArray[1, {#, #}] - IdentityMatrix[#]][
      "NonzeroPositions"], # - 1] &[n] // RepeatedTiming // First

Array[List, {n, n}, {1, 1}, DeleteCases[{##}, {a_, a_}, {2}] &] // 
  RepeatedTiming // First

Pick[a, IdentityMatrix[n], 0] // RepeatedTiming // First

0.00833

0.419

0.560

0.2670

0.491

0.00460

0.0351

0.412

0.0274

So even including the time to build a my Delete approach is considerably faster than any other method yet proposed. Second fastest is kglr's SparseArray, or Pick inspired by it.

Partition[#, 3]&@Permutations[Range@4, {2}] // MatrixForm

Or

Select[#, DuplicateFreeQ] & /@ Outer[List, Range@4, Range@4]