Example 4 Given a positive integer $n>1$. Let $a_{1}, a_{2}, \cdots, a_{n}$ be a permutation of $1,2, \cdots, n$. If $i<j$ and $a_{i}<a_{j}$, then $(a_{i}, a_{j})$ is called an ordered pair. $X$ is the number of ordered pairs in $a_{1}, a_{2}, \cdots, a_{n}$. Try to find $E(X)$.
(2020, University of Science and Technology of China Innovative Class Preliminary Exam)