I haven't checked for efficiency of evaluation, but for sheer simplicity of code, why not just the following?

x[i_, j_] := const Log[Max[i, j] h]
x[i_, i_] := const Log[0.44705 h]
Table[x[i, j], {i, 1, 10}, {j, 1, 10}] // MatrixForm

That uses the symmetry of the condition for different i and j, and it exploits the fact that a specific rule, here

x[i_,i_]:= 

always overrides a general rule, here:

 x[i_,j_]:=

Hi Sergio,

I propose you to verify the following code with Table[]:

    mo = 4/10  \[Pi];
    \[Sigma] = 3533568905/100;
    w = 120 \[Pi];
    h = 5/1000;
    const = (w  mo \[Sigma] h^2)/(2. \[Pi]);

    x = Table[
       Which[i < j, const Log[j h],
             i == j, const Log[0.44705 h],
             i > j, const Log[i h]], 
       {i, 10}, {j, 10}];
    x//MatrixForm

Your matrix is defined as a list $x$. Element $(i,j)$ of the matrix $x$ can be addressed by applying the syntax $x[[i,j]]$.

http://reference.wolfram.com/language/ref/Do.html.en

Hi Sergio, this should work. Please check carefully against the results of the old code

xm = Array[x, {10, 10}];
For[i = 1, i <= 10, i++,
 xi = (N[i] - 0.5) h;
 x[i, i] = const * Log[0.44705*h*xi*2.];
 For[j = 1, j <= 10, j++,
  If[i == j, Break[]];
  xj = (N[j] - 0.5) h;
  x[i, j] = const* Log[Abs[xi - xj]* Abs[xi + xj]];
  x[j, i] = x[i, j];
  ]
 ]

Here you can see the structure of the matrix before putting in the numbers

MatrixForm[xm]

Finally the variables are replaced by the numeric values

prms = {g -> 1.*10^8 / 2.83, freq -> 377., h -> 5.*10 (-3), 
   perm -> 4.* \[Pi]* 10^(-7), eo -> 1.};
MatrixForm[xm] /. const -> (freq * perm* g * h^2) / (2. \[Pi]) /. prms