It's well known that there is an equivalence between the category of $2$-dimensional topological quantum field theories (TQFTs) over $\mathbb{C}$ and commutative Frobenius $\mathbb{C}$-algebras.
This is fairly easy to prove by some standard diagram manipulation and a bit of topological intuition.
Given that higher dimensional TQFTs are much more complicated than their $2$-dimensional cousins, I'm not expecting a full answer, but hopefully some partial results.
Is there an algebraic category which is equivalent to the category of $n$-dimensional TQFTs over $\mathbb{C}$?
It's possible that the 'algebraic category' part of the above question isn't well-formulated enough, but I hope that at least the spirit of my question is clear. Can we study the collection of $n$-dimensional TQFTs in an 'algebraic way'?
I think either the answer is no, or the question is too hard in this form. From what I can tell, looking for an algebraic way to encode higher dimensional TQFTs is what inspired people to start looking at extended TQFTs which are more naturally encoded algebraically. For references see Baez-Dolans important paper here and Lurie's paper sketching a proof of Baez and Dolan's conjecture (which is in some sense an analogue of the 2d case for higher TQFTs) here.
For 2d TQFTs, the really good thing that happens is that the category of 2-bordisms has a really nice description. It is generated by the circle $S^1$ along with the following relations (credit for the picture goes to Chris Schommer-Pries thesis here):
along with the cylinder which corresponds to the identity morphism. This makes the 2 dimensional bordism category the free monoidal symmetric category on one generator. The statement that 2d TQFTs are classified by Frobenius algebras is really this description of bordism category along with the fact that these generators and relations correspond to the algebraic structure of a Frobenius algebra.
In general, if we can give a similar generator and relation description of the $n$-bordism category, then it would be equivalent to whatever algebraic structure those generators and relations define. However, (see page 8 of Lurie's paper) when we get into higher dimensions, it is not in general going to be possible to break up any $n$-manifold into simple pieces like we could with $2$-manifolds into discs, cylinders, and pairs of pants.
This leads naturally to the notion of extended TQFT instead. You can look at those links for the exact definitions which involves higher categories and such, but the idea is to not just look at $n$-manifolds as morphisms between $n-1$ manifolds but also look at $n-1$ manifolds as morphisms between $n-2$ manifolds, etc. This gives the structure of an $n$-category. An $n$-dimensional extended TQFT would then be a functor from this category to an appropriate $n$-category of vector spaces. Then looking at these extended TQFTs, the analogous statement to the simple generators and relations of the bordism category for $2$ dimensions is the Baez-Dolan Cobordism Hypothesis, which in one formulation states that the $n$-category of extended $n$-dimensional bordisms is the free symmetric monoidal $n$-category on one object. Then extended $n$-dimensional TQFTs would be classified by whatever algebraic object corresponds to the relations on a free symmetric monoidal $n$-category. This is a lot of data that I haven't looked at carefully so I don't know exactly what this is or if this even has a name. For more details and some real math look at those three links I gave.