What is the right definition of the n dimensional Hodge dual and cross product?

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Could you help me with those geometric algebra confusions:

Which one is the right definiton of pseudovector ? $$(n-1)vector\\ \text{or}\\ *((n-1)vector)$$

I found one can express the hodge dual this way: $$*((k)vector)=pseudo (n-k)vector\\*(pseudo (k)vector)=(n-k)vectors$$ But from the definition "$*:\wedge^kV\rightarrow\wedge^{n-k}V$", I don't see where pseudos come from. I think it's from $sgn(\sigma)$ or something like that

Can the link between cross product and wedge product be generalized to other dimensions using: $$\times:\wedge^rV\times\wedge^{s-r}\rightarrow \wedge^{n-s}V$$ $vector\times vector$ would then give a pseudoscalar in 2D and a pseudobivector in 4D