Suppose a vector is like this in $x$-$y$ plane.
Now suppose a question is asked what is the angle betweeen vector $B$ and $A$. What will be the answer? Is it $\theta$ or $360-\theta$ ? I am asking this question because I have heard two contradictory rules for this.
First rule: It should be taken in anti-clockwise direction.
Second rule: The smaller angle should be taken.
Notice that the big angle plus the small angle is $2\pi$ radians and: $$ sin(2\pi-x)=-sin(x) $$
Notice that the vector that is the cross-product of $\overrightarrow{A}$ and $\overrightarrow{B}$ is perpendicular to both $\overrightarrow{A}$ and $\overrightarrow{B}$. If you see, right-hand-rule, it is clear that the answer depends on whether you are looking for $\overrightarrow{A} \ X\ \overrightarrow{B}$ or $\overrightarrow{B} \ X\ \overrightarrow{A}$
Notice that by property of cross product: $$ \overrightarrow{A} \ X\ \overrightarrow{B}=-(\overrightarrow{B} \ X\ \overrightarrow{A}) $$
Hence the answer to your question depends:
If you are looking for $\overrightarrow{A} \ X\ \overrightarrow{B}$ then, use the angle that goes counter-clockwise from $\overrightarrow{A}$ to $\overrightarrow{B}$. If you are looking for $\overrightarrow{B} \ X\ \overrightarrow{A}$ then, use the angle that goes counterclockwise from $\overrightarrow{B}$ to $\overrightarrow{A}$.
Edit: as mentioned by many, there are plenty of resources that explained this better, if you just type "right-hand-rule" on google