Show that $x,y \text{ and } z$ are linear independent iff $x \times y, x \times z \text{ and } y \times z$ are linear independent

Let $x,y,z \in{\mathbb R}^3$. Show that $x,y \text{ and } z$ are linear independent iff $x \times y, x \times z \text{ and } y \times z$ are linear independent. Where $\times$ denotes the cross product in ${\mathbb R}^3.$

So far I have only been able to show the forward implication: If $x,y,z$ are linear dependent, then $x \times y, x \times z \text{ and } y \times z$ are also linear dependent.

I need help with the backwards implication: If $x \times y, x \times z \text{ and } y \times z$ are linear independent, then $x,y,z$ are linear independent

I would be grateful for some help.

Assume the cross products are linearly independent. Suppose $ax+by+cz=0$ so $ax\times y + cz\times y =0$. Hence $a=c=0$. Similarly you can prove $b=0$. Since no non-zero choices of $a,\,b,\,c$ work, $x,\,y,\,z$ are linearly independent.