V is a real $n$-dimensional vector space. I'm using the notation as in John Lee's Riemannian Manifolds: an Introduction to Curvature for the set of $\binom{k}{l}$ tensors and the set of multilinear mappings, which appear respectively on the title. Please note that $(V)^l = V\times ...\times V$ $l$ times.
I've tried doing it by first noting that they are both real vector spaces of the same dimension and building the following homomorphism $$ \phi: T^k_{l+1}(V)\rightarrow \mathcal{L}\left( (V^*)^k \times (V)^l\right) $$ which maps $A$ into $\phi(A)$. $\phi(A)$ is the multilinear mapping in $\mathcal{L}\left( (V^*)^k \times (V)^l\right)$ that should send $(w_1,...,w_k,x_1,...,x_l)$ into a vector of $V$. Here, $w_i\in V^*$ and $x_j\in V$ for all $i=1,...,k$ and $j=1,...,l$.
If I write it as $\phi(A)=A(w_1,...,w_k,x_1,...,x_l,x_{l+1})$, then I have 1 vector untouched on its argument (namely $x_{l+1}$), but that does not mean that $\phi(A)$ is a vector and, should it be, I can't say that the vector $\phi(A)$ depends on the other variables.
I wish I could get some help in this final step. Can I say that $T^k_{l+1}(V)$ is equal to the set of endomorphisms of some vector space? If I could then I would write $A$ in matrix form and try to figure out the rest.