Is geometric algebra (Clifford algebra) isomorphic to tensor algebra? If so, how then would one relate a unique 2-vector (this is what I'm going to call a multivector that is the sum of a scalar, vector, and bivector) to every 2nd rank tensor?

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*Edit by the OP, converted from "answer"*

Okay. Well I'm still curious if there's a way to represent any 2nd-rank tensor by a bivector, vector, and scalar. Or in particular, can any $3 \times 3$ matrix be represented by a 2-vector in 3D.

It seems to me that they can't because I would guess the matrix representation of a bivector (grade 2 element of a 2-vector) would be exactly the same as the $3 \times 3$ matrix representation of the cross product (i.e. $[a \times b]\_{ij} = a\_j b\_i - a\_i b\_j$) which only uniquely identifies 3 components.  

I would also assume that the scalar part of the 2-vector would be represented by a scalar times the $3 \times 3$ identity matrix. This would fill in 3 numbers, but really only uniquely gives 1 component.  

I don't know how to represent the vector component of the 2-vector as a $3 \times 3$ matrix but I don't see how it could identity the remaining 5 components by itself.

Am I right then in assuming that there is a canonical matrix representation of a general 2-vector, but that there are matrices that cannot be represented by any 2-vector?