No.
The vector cross product takes two vector operands to produce a vector result. The result, like all geometric vectors, has two properties: length and orientation.
If u and v are vectors in three dimensional space, then u × v is a three dimensional vector.
Definition of the Cross product u × v
To find a vector perpendicular to a particular plane, compute the cross product of two vectors in that plane. But there are two directions perpendicular to the plane. Which one does the cross product give you? That is determined by the right hand rule, which will be explained on the next page.
A wood screw is positioned perpendicular to a plank of wood. You wish drive the screw into the wood with a screw driver. Which direction must you twist the screw driver?