计算下列 $n$ 阶行列式.
(1)
\[
\begin{vmatrix}
1 & 2 & 3 & \cdots & n-1 & n\\
2 & 3 & 4 & \cdots & n & n+1\\
3 & 4 & 5 & \cdots & n+1 & n+2\\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots\\
n & n+1 & n+2 & \cdots & 2n-2 & 2n-1\\
\end{vmatrix}
\]