Is it possible to “take out” the cross product of unit vectors to calculate separately?

Let's say I have two different vectors, which are $$\vec v_1=Ae^{i(kz-wt)}\hat y$$ and $$\hat v_2 =Be^{i(kz-wt)}$$ and I want to take the cross product of the two of them, can I say that the following is true?

$$\vec v_1 \times \vec v_2=\hat x\ \times \hat y \ (v_1v_2)$$

I've been trying to figure this out on my own but I haven't been able to. I was told this is true but it doesn't make sense to me because it doesn't seem to hold when I use the definition of the cross product using it's sine form.

Using the definition you've alluded to at the end of your post, $$\vec{v_1} \times \vec{v_2} = \|\vec{v_1}\| \|\vec{v_2}\| \sin(\theta) \vec{n} = \|\vec{v_1}\| \|\vec{v_2}\| (\hat{x} \times \hat{y})$$ where $$\hat{x} = \vec{v_1}/\|\vec{v_1}\|$$ and $$\hat{y} = \vec{v_2}/\|\vec{v_2}\|$$, where $$\theta$$ is the angle between $$\vec{v_1}$$ and $$\vec{v_2}$$, and where $$\vec{n}$$ is the normal vector to $$\vec{v_1}$$ and $$\vec{v_2}$$.