How can a Cross-Product give a vector solutions is missing [closed]

I am trying to understand the function of a cross-product

If you take the cross-product of V x U gives a another vector that is pendicular say call it c-vector?

V * c-vector = 0

U * c-vector = 0

But what i don't understand why can't this vector be written as linear combinations of

aV + bU = c-vector ?

And another question is the Cross-product used to find basis vectors?

The cross product is defined for vectors in $\mathbb{R}^3$.

If the two vectors $U$ and $V$ are linearly independent they determine a plane through the origin. Then their cross product is perpendicular to both, so perpendicular to the plane, so clearly not a linear combination.

If the two vectors are not linearly independent then their cross product is the $0$ vector and is a linear combination.

You should ask your "other question" as another question, with more context.