计算 $n$ 阶行列式：

$$
D=\left|\begin{array}{ccccccc}
1 & 2 & 3 & \cdots & (n-2) & (n-1) & n \\
1 & -1 & 0 & \cdots & 0 & 0 & 0 \\
0 & 2 & -2 & \cdots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & (n-2) & -(n-2) & 0 \\
0 & 0 & 0 & \cdots & 0 & (n-1) & -(n-1)
\end{array}\right|
$$