Example 1 Connect any two vertices of a cube with a straight line, among these lines, the number of pairs of mutually perpendicular skew lines is $\qquad$.
Solution: Fill in 78. Reason: Because there are 6 lines (4 edges and 2 face diagonals) that are perpendicular and skew to one edge, 12 edges can generate $12 \times 6=72$ pairs of skew lines.

Secondly, there are 5 lines (2 edges, 1 face diagonal, and 2 body diagonals) that are perpendicular and skew to one face diagonal, 12 face diagonals can generate $12 \times 5=60$ pairs of skew lines.

Finally, there are 6 lines (6 face diagonals) that are perpendicular and skew to one body diagonal, 4 body diagonals can generate $4 \times 6=24$ pairs of skew lines.

Since each pair of skew lines is counted twice in the above counts, the total number of mutually perpendicular skew lines is $\frac{1}{2}(72+60+24)=78$ pairs.