The angular momentum operator swaps directions. There are 3 distinct components of angular momentum operator in 3 dimensions and 6 components in 4 dimensions. In general, the whole angular momentum operator is an antisymmetric matrix, numbered by the dimensions, so it has as much components.
Regarding your "matrix elements" in the sense of position representation. First pick some specific component of the angular momentum operator. I.e.
J_{xy} - swaps the direction x with direction y.
Now you want to calculate
<a|J_{xy}|b>
where |a> and |b> are position eigenvectors, i.e. Dirac deltas \delta(q - a) and \delta(q - b) where q is the position parameter. (We can't use characters x and y here, since we have reserved them for direction indicators.)
As we said, what the angular momentum operator does with the position eigenvector is to swap it with some other direction, i.e. rotate by 90 degrees. So:
J_{xy}|x> = |y>
J_{xy}|y> = -|x>
Here, |x>, |y> are the unit position eigenvectors pointing in x and y directions, respectively. I.e. they have position components: [1, 0, 0] and [0, 1, 0]. Remember, x and y are not variables, they are just labels.
More generally, the angular momentum rotates a vector by 90 deg.
J_{xy}|a> = R(z, 90)|a>
Here, the symbol R(z, 90) means "rotation matrix that rotates by 90 degrees over the axis pointing in the z direction". The z direction is the direction perpendicular both to x and y. It is always so in the case of rotations. In 3 dimensions you can think of it as the cross product of x and y. In higher dimensions, it is a bit more complex.
Going back to your original problem, the "matrix elements" in the position representation will be something like:
<a|J_{xy}|b> = <a|R(z, 90)|b> = \delta^3(a - R(z, 90)b)
Here, a and b are variables ranging over 3-dimensional position tuples. The second one is rotated by 90 degrees. I hope you understand.
Note that you have more angular momentum operators, J_{xz} and J_{yz} plus their linear combinations. You can construct an angular momentum operator for any direction and any angle this way. The "matrix elements" will be computed the same way in these cases, with the necessary modifications (rotation direction and angle).
