Skyscraper sheaf in a s.e.s On a curve $C$, if $\mathbb{C}_p$ is the skyscraper sheaf at a point $p \in C$, then we have the exact sequence $0 \to \mathcal{L}(-p) \to \mathcal{O}_C \to \mathbb{C}_p \to 0$. On a variety with arbitrary dimension, how can I put $\mathbb{C}_p$ in a s.e.s in a way that defines $\mathbb{C}_p$ considering that $\mathcal{L}(p)$ might not be a line bundle? One could write $0 \to \mathcal{I} \to \mathcal{O}_X \to \mathcal{O}_X \otimes \mathbb{C}_p \to 0$, but then what is $\mathcal{I}$ really? It doesn't look like a line bundle for some arbitrary variety $X$. Thanks.

If $X =\mathbb C^n$ with coordinates $z_1,\dots,z_n$ and $p=(0,\dots,0)$ write $$0\to z_1\mathcal O_X+\dots+z_n\mathcal O_X\hookrightarrow \mathcal O_X\to \mathbb{C}_p\to 0\quad (\bigstar)$$  
If $X$ is an arbitrary manifold of dimension $n$ write $$0\to \mathcal I_p\hookrightarrow \mathcal O_X\to \mathbb{C}_p\to 0 \quad (\bigstar \bigstar)$$ where $\mathcal I_p$ is the ideal subsheaf of $\mathcal O_X$ of functions vanishing at $p$.
Locally at $p$ that subsheaf $\mathcal I_p$ looks like $z_1\mathcal O_X+\dots+z_n\mathcal O_X$ in $(\bigstar)$ while on  $U=X\setminus \{p\}$ we have $\mathcal I_p|U=\mathcal O_X|U$.
The sheaf $\mathcal I_p$ is always coherent, locally free for $n=1$  but never locally free for $n\geq 2$ .
