TITLE: Tensors as matrices vs. Tensors as multi-linear maps
QUESTION [10 upvotes]: So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related.  For instance, it seems that a (1,1)-tensor should be able to be realized as a 2 by 2 matrix, as something that looks like $(x_1,x_2)\otimes(y_1,y_2)$ and as a multi-linear map from $\mathbb{R}^2\times(\mathbb{R}^2)^\ast$ to $\mathbb{R}$.  Is there a formal isomorphism between these three ideas?  
My motivation for asking this is that I'm trying to work with deRham cohomology without just doing the mechanics of it.  I'd like to actually know what's going on, and so I'm trying to see 
1) How is $\mathrm{Alt}^p(\mathbb{R}^n)$ actually the same as $\wedge^p(\mathbb{R}^n)$ where the former is the space of alternating p-linear maps and the latter is the algebraically defined quotient of $\mathbb{R}^n\otimes\ldots\otimes\mathbb{R}^n$, the tensor product of $p$ copies of $\mathbb{R}^n$?
2) How is $\Omega^p(\mathbb{R}^n)$ actually the dual of p-vector fields (if in fact it is...)?
Thanks!

REPLY [5 votes]: This interpretation makes judicious use of duality.  Recall that $Hom_R(A\otimes B, C)\cong Hom_R(A,Hom_R(B,C))$ (the proof of this fact is extremely simple.  Please give it a try.).  With finite dimensional vector spaces, we have a canonical isomorphism $V\cong V^{**}=Hom_R(Hom_R(V,R),R)$.  Then to give a map $A\to B$, this is the same as giving a map $A\to B^{**}$ by composition, and by the earlier note about the hom-tensor adjunction, this is identical to giving a map $A\otimes B^*\to R$.  
With regards to your two questions, they fall out from these elementary observations.  I urge you to try to work them out.