Example 1 If three lines $a$, $b$, and $c$ in space are pairwise skew lines, then the number of lines that intersect with lines $a$, $b$, and $c$ is ( ).
(A)0
(B) 1
(C) more than 1 but a finite number
(D) infinitely many
(1997, National High School Mathematics League Competition)
Solution: First, regardless of the positional relationship between lines $a$, $b$, and $c$, we can always construct a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ such that $A B$ lies on line $a$, $B_{1} C_{1}$ lies on line $b$, and $D D_{1}$ lies on line $c$. Next, take any point $M$ on the extension of $D D_{1}$, and determine a plane $\alpha$ by point $M$ and line $a$. Plane $\alpha$ intersects line $B_{1} C_{1}$ at point $P$ and line $A_{1} D_{1}$ at point $Q$, so $P Q / / a$, as shown in Figure 1. Thus, within plane $\alpha$, line $P M$ does not parallel line $a$, and $P M$ must intersect $a$ at a point $N$. Therefore, line $M N$ intersects lines $a$, $b$, and $c$ simultaneously. Since there are infinitely many ways to choose point $M$, there are infinitely many lines that intersect lines $a$, $b$, and $c$ simultaneously. Hence, the answer is (D).