Clear[n, substitutions];
substitutions[n_] :=
   {{1, 1} -> 1,
    {1, 2} -> 0,
    {1, 3} -> -1,
    {n, n - 2} -> 1,
    {n, n - 1} -> 0,
    {n, n} -> -1};
conditionnumbers =
  Table[{i, (Last[
       mydone =
        Sort@N@Abs@
           Eigenvalues[
            ReplacePart[basicMatrix[i] , substitutions[i]]]]/
      First[mydone])}, {i, 4, 40}];
ListPlot[conditionnumbers]