My problem is perhaps a general lack of understanding but it occurred in a special case of a theorem in Eisenbud's and Harris' "The geometry of schemes" (Theorem VI-29). Let $K$ be a field and $n\in\mathbb{N}$.

Given a closed subscheme $X$ of $\mathbb{P}^n$ with Hilbert Polynomial $P$. The tangent space $T_{[X]}$ to the Hilbert scheme $\mathcal{H}_{P}$ at the point corresponding to $X$ is the space of global sections of the normal sheaf $\mathcal{N}_{\mathbb{P}^n/X}$.

I don't know what "is" precisely means here. $T_{[X]}$ has a natural $K([X])$-vector space structure and I would like to have an isomorphism of $K([X])$-vector spaces but my problem begins much earlier in the definition of $\mathcal{H}om$-sheaves:

Let $J$ be the ideal sheaf of $X$. The presheaf $$ U \mapsto \mathcal{H}om_{\mathcal{O}_X|_U}(J/J^2|_U,\mathcal{O}_X|_U) $$ is already a sheaf of $\mathcal{O}_X$-modules (this does not depend on the special arguments here). This sheaf $\mathcal{N}_{\mathbb{P}^n/X}$ is called the normal sheaf of $X$ in $\mathbb{P}^n$. The global sections $$ \mathcal{N}_{\mathbb{P}^n/X}(X)=\mathcal{H}om_{\mathcal{O}_X}(J/J^2|_X,\mathcal{O}_X) $$ of this normal sheaf is an $\mathcal{O}_X(X)$-module. Is there an isomorphism of $\mathcal{N}_{\mathbb{P}^n/X}(X)$ to some $hom$ of rings or modules where I can work with? What is a possible $K([X])$-vector space structure on it?