Can a vector field always be written as a cross-product of two other vector fields? As the title suggests, can any arbitrary conservative vector field, $\bf{F}$ $= \langle P,Q,R \rangle$, where the component functions are functions of $(x,y,z)$, always be written as a cross-product of two other vector fields, namely $\bf{A} \times \bf{B}$? Going further, let $\bf{A}$ $= \nabla a$, where $a = a(x,y,z)$ and has continuous first partial derivatives. Also, let $\bf{B}$ $=\langle b_1,b_2,b_3\rangle$ where the component functions are functions of $(x,y,z)$.
What I have so far is the following:
$\cases{
P=\frac{\partial a}{\partial y}b_3-\frac{\partial a}{\partial z}b_2\\
\\
Q=\frac{\partial a}{\partial z}b_1-\frac{\partial a}{\partial x}b_3\\
\\
R=\frac{\partial a}{\partial x}b_2-\frac{\partial a}{\partial y}b_1
}$
The reason I am asking is that I recently solved the following question:
Let $f$ be a smooth function defined on $\mathbb{R}^3$. Let $S$ be the level surface, $\{(x,y,z):f(x,y,z)=c\}$ for some $c \in \mathbb{R}$. Assume $\nabla f$ is never the zero vector on $S$ and let $\bf{F}$ $=\nabla f$. Show that the surface integral over $S$ of $\bf{F} \times \bf{G}$ is $0$ for any $\bf{G}$. ($\star$)
And I thought I could use this property if I can take a complicated surface integral and attempt to break it down to this case to easily see answers that are zero. So I suppose I am asking for the conditions that arbitrary vector fields need to satisfy to use the result of the property in the above problem.
Thanks!
$\bf{EDIT}$
Summary of questions:
Q1. Can a vector field always be written as a cross-product of two other vector fields? $\bf{Answered.}$
Q2. Given that $\bf{F}$ $=\nabla f$ can be written as a cross-product of two vector fields, what other conditions are necessary to use the result of $(\star)$?    $\bf{Answered.}$

Not continuously, no. If the vector field $\mathbf F(\mathbf x)=\mathbf x$ could be written as $\mathbf A\times\mathbf B$, then $\mathbf A$ restricted to the sphere $S^2$ would be a nonvanishing tangent vector field, contradicting the hairy ball theorem.
