The following is Hartshonre's definition of geometric vector bundles:
Let $Y$ be a scheme. A (geometric) vector bundle of rank $n$ over $Y$ is a scheme $X$ and a morphism $f:X\to Y$ together with additional data consisting of an open covering $\{U_i\}$ of $Y$ and isomorphisms $\psi_i:f^{-1}(U_i)\to \mathbb A^{n}_{U_i}$ such that for any $i,j$, and for any open affine subset $V=\operatorname{Spec}A \subseteq U_i\cap U_j$ the automorphism $\psi = \psi_j\circ\psi_{i}^{-1}$ of $\mathbb A^{n}_{V}$ is given by a linear automorphism $\theta$ of $A[x_1,\cdots,x_n]$, i.e, $\theta(a)=a$ for any $a\in A$ and $\theta(x_i)=\sum a_{ij}x_j$ for suitable $a_{ij}\in A$.
It is easy to show that a vector bundle of rank $n$ is equivalent to the data $\{U_i,\psi_{ij}\}$, where $\{U_i\}$ is an open covering of $Y$, $\psi_{ij}:\mathbb{A}_{U_i\cap U_j}^n \to\mathbb{A}_{U_j\cap U_i}^n$ are isomorphisms satisfying cocycle condition (i.e. $\psi_{ik}=\psi_{jk}\circ \psi_{ij}$), which are induced by linear automorphisms when restricted to open affine subsets $V\subseteq U_i\cap U_j$.
My question is :
Is a vector bundle of rank $n$ equivalent to the data $\{U_i,\Psi_{ij}\}$, where $\{U_i\}$ is an open covering of $Y$, and $\Psi_{ij}\in GL_n(\mathcal{O}(U_i\cap U_j))$ satisfying cocycle condition?
I know that $\Psi_{ij}\in GL_n(\mathcal{O}(U_i\cap U_j))$ give us morphisms $T_{ij}: U_i\cap U_j \to GL_n$ which satisfy cocycle condition. And in the manifold case, a vector bundle is equivalent to the data $\{U_i,T_{ij}\}$. So I wonder if it still holds in algebraic geometry.