Example 1 Draw a line through any 2 vertices of a cube. Among these lines, the number of pairs of skew lines that are not perpendicular to each other is $\qquad$.
(2000 Hunan Province Middle School Mathematics Olympiad Summer Camp Question)
First, there are 6 lines (4 face diagonals and 2 body diagonals) that are neither perpendicular nor coplanar with a given edge. Therefore, 12 edges can generate $12 \times 6 = 72$ pairs of skew lines.

Second, there are 8 lines (4 edges and 4 face diagonals) that are neither perpendicular nor coplanar with a given face diagonal. Therefore, 12 face diagonals can generate $12 \times 8 = 96$ pairs of skew lines.

Finally, there are 6 lines (6 edges) that are neither perpendicular nor coplanar with a given body diagonal. Therefore, 4 body diagonals can generate $6 \times 4 = 24$ pairs of skew lines.

Since each pair of skew lines is counted twice in the above calculations, the total number of non-perpendicular skew lines is $\frac{1}{2}(72 + 96 + 24) = 96$ pairs.