Short question
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Can we describe a quasi-coherent module on a scheme by usual modules with respect to an affine cover, which satisfy some compatibility conditions, which can be formulated in the language of commutative algebra (actually tensor products of modules)? Thus, restrictions to non-affine subsets are not allwed. 


Example
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If the scheme is separated, the answer is of course yes: Choose an affine cover $U\_i$ of our scheme $X$, thus also $U\_i \cap U\_j$ is affine. If $M\_i$ is a $\mathcal{O}(U\_i)$-module such that we have isomorphisms $\phi\_{ij} : M\_i \otimes\_{\mathcal{O}(U\_i)} \mathcal{O}(U\_i \cap U\_j) \cong M\_j \otimes\_{\mathcal{O}(U\_j)} \mathcal{O}(U\_i \cap U\_j)$ of $\mathcal{O}(U\_i \cap U\_j)$-modules, which satisfy the cocycle conditions, namely $\phi\_{ii} = id$ and $\phi\_{ijk} = \phi\_{kji} \phi\_{ikj}$ where $\phi\_{ijk} = \phi\_{ij} \otimes\_{\mathcal{O}(U\_i \cap U\_j)} \mathcal{O}(U\_i \cap U\_j \cap U\_k)$. Then the quasi-coherent modules $\widetilde{M\_i}$ on $U\_i$ can be glued to a quasi-coherent module on $X$.


Long question
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In the general case, there is an affine cover $U\_i \cap U\_j = \cup\_k W\_k$, such that $W\_k$ is basic-open in $U\_i$ and $U\_j$ simulteneously. In this setting we should use isomorphisms $\phi\_{ijk} : M\_i \otimes\_{\mathcal{O}(U\_i)} \mathcal{O}(W\_k) \cong M\_j \otimes\_{\mathcal{O}(U\_j)} \mathcal{O}(W\_k)$. However, in order to formulate the cocycle condition in affine terms, we actually have to cover triple overlaps $U\_i \cap U\_j \cap U\_k$ by affines $R\_n$, such that $R\_n$ is basic-open in $U\_i,U\_j,U\_j$ and then cover the $U\_i \cap U\_j$ with these $R\_n$, where we vary $k$. Thus we assume isomorphisms $\phi\_{ijkn} : M\_i \otimes\_{\mathcal{O}(U\_i)} \mathcal{O}(R\_n) \cong M\_j \otimes\_{\mathcal{O}(U\_j)} \mathcal{O}(R\_n)$ and may formulate the cocycle condition. Now in order to glue these isomorphisms to an isomorphism on $U\_i \cap U\_j$ between $\widetilde{M\_i}$ and $\widetilde{M\_j}$, we also have to ensure that there is some intersection compatibility when we vary $k$. However, this cannot be formulated with the given data.


Before I end up defining refinements of these affine covers by transfinite induction ;-), I thought it would be better to ask here if there is an easy way which I overlook.



Quasi-coherent module given by modules and compatibility conditions in the language of commutative algebra
I think the explicit description that you suggest can be wrapped up as follows.


For every $i,j$, let $C\_{ij}$ be a family of indexes such that 
$$U\_i\cap U\_j=\bigcup\_{a\in C\_{ij}} W\_a,$$
with $W\_a$ being basic open in $U\_i$ and in $U\_j$. 
We may assume that $C\_{ii}$ is a singleton set and $C\_{ij}=C\_{ji}$.
Let $\phi\_{ij,a}$ be as in your question, $a\in C\_{ij}$. Impose the cocycle condition 
$$\phi\_{ij,a}\circ\phi\_{jk,b}=\phi\_{ik,c}$$
on $W\_a\cap W\_b\cap W\_c$ for all $a\in C\_{ij}$, $b\in C\_{jk}$, $c\in C\_{ik}$. Notice that
$W\_a\cap W\_b\cap W\_c$ is a basic open in $W\_a$, $W\_b$, $W\_c$, and also in $U\_i$, $U\_j$, $U\_k$,
so this is an identity in algebraic terms, as required.


A more natural description:


A quasicoherent sheaf on $X$ is given by a $\Gamma(U,O)$-module $M\_U$ for every affine open $U\subset X$ and an isomorphism $\phi\_{U,V}:M\_U|\_V\to M\_V$ whenever $V\subset U$ is a basic
open, such that $\phi\_{U,V}$ satisfies obvious cocycle condition on triples $W\subset V\subset U$.


