Cross-products and the right-hand rule

I understand how to do problems relating to this, I just don’t really understand the concept behind it.

Why is the torque vector in rotational motion directed out of the plane of rotation? I can’t wrap my head around why it would be directed that way in a real-world set of mind.

What does a cross-product actually represent? I understand that its magnitude is the area of a parallelogram formed by the two crossed vectors. I’m not sure why the determinant of a 3×3 matrix actually means something however. I know what it means in terms of mathematics, but not in terms of real-world forces and physics that I can conceptualize.

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