I have been looking EVERYWHERE for just one actual example of a wedge product being calculated and for a bi-, tri- or any multivector to be written out with actual numbers!

The reason why this is the case is because the wedge product of two vectors is not just some array of numbers. Sure, every finite dimensional vector space is isomorphic to the vector space of the tuples of that length, hence you can describe it as array of numbers (by mapping it with that isomorphism which simply maps the basis of your space to some basis of the tuples). However is that meaningful thing to do? In some cases it might be but in general you have to realize that this is just an arbitrary choice. There are a lot of bases so you can do this in many ways.

In the three dimensional case there's important map, the Hodge isomorphism, that gives you a way to represent the wedge of two vectors with a vector from your original space (this is actually the cross product, if you chose the right parameters for the map), however this map requires two extra parameters that introduce more structure to your space (namely inner product and a choice of volume form).

In the case of $(1, 3, -2) \wedge (5, 2, 8)$, there's not much else that you can do other than picking an arbitrary basis of your original space and expressing that product in terms of the wedged basis vectors. For example
$$(1,3,-2) = 1(1, 0, 0) + 3(0,1,0) -2(0,0,1)$$
$$ (5,2,8) = 5(1, 0, 0) + 2(0,1,0) +8(0,0,1) $$
$$ (1,3,-2) \wedge (5,2,8) = (1(1, 0, 0) + 3(0,1,0) -2(0,0,1))
\wedge (5(1, 0, 0) + 2(0,1,0) +8(0,0,1))
=
5 (1, 0, 0) \wedge (1, 0, 0) + 15 (0,1,0) \wedge (1, 0, 0) - 10 (0,0,1) \wedge (1, 0, 0) + 2 (1, 0, 0) \wedge (0,1,0) - 4 (0,0,1) \wedge (0,1,0) +
8 (1, 0, 0) \wedge (0,0,1) + 24 (0,1,0) \wedge (0,0,1) - 16 (0,0,1) \wedge (0,0,1)
=
-15 (1, 0, 0) \wedge (0,1,0) + 2 (1, 0, 0) \wedge (0,1,0) + 10 (1, 0, 0) \wedge (0,0,1) + 8 (1, 0, 0) \wedge (0,0,1) + 24(0,1,0) \wedge (0,0,1) +
4 (0,1,0) \wedge (0,0,1)
=
28 (0,1,0) \wedge (0,0,1) + 18 (1, 0, 0) \wedge (0,0,1) - 13 (1, 0, 0) \wedge (0,1,0)
$$

Or now using the mentioned arbitrary isomorphism (mapping that chosen basis to the chosen basis of tuples) you can map it to just array of numbers hence (28,18, -13) but that's not too meaningful because it doesn't explicitly show the basis.