Example 8 Let $n$ be a given natural number, and $n \geqslant 3$, for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, denote $\left|a_{i}-a_{j}\right|$ $(1 \leqslant i<j \leqslant n)$ the minimum value as $m$. Find the maximum value of the above $m$ under the condition $a_{1}^{2}+a_{2}^{2}+$ $\cdots+a_{n}^{2}=1$.