Let $E$ be a holomorphic $m$-dimensional vector bundle over complex (analytic) space $X$ with projection $p:E \to X$. We now want to endow $E$ with sheaf structure $\mathcal{O}_E$ as follows:
We set
$$\Gamma(U,\mathcal{O}_E):= \{s : U → E \vert _U \text{ holomorphic section, therefore we have } p \circ s = id_U \}$$
My question is how is concretely defined a holomorphic function $s: U \to E$?
If $X$ is a complex $n$-manifold then you can concretely $s$ locally realize on $\mathbb{C}^n \cong V \subset U$ as restriction $$s \vert _V = (f_1, ..., f_m, id_1, ..., id_n): V \to E \vert V \cong \mathbb{C}^m \times V = \mathbb{C}^m \times \mathbb{C}^n$$
for holomorphic functions $f_i: \mathbb{C}^n \to \mathbb{C}$ in sense of basic complex analysis (as polynomial series).
But if $X$ not a complex manifold but more generally a complex space (for example complex analytical space) how the holomorphic functions $s: U \to E$ are defined / described?
Which local shape do they have?