We're learning about this in chemistry right now. What happens in 3D goes like this:

2 pairs align themselves along a line

3 pairs point at the corners of an equilateral triangle

4 pairs point at the corners of a regular tetrahedron

5 pairs have the equilateral triangle plus the line, perpendicular to each other

6 pairs have three lines, each perpendicular to each other

Now what I'd like to know is if you can do more than 6 pairs in 3D, and what happens in 4D, after 5 pairs (2-4 are the same, 5 is obviously going to point at the corners of a regular pentachoron).