Obviously, a bilinear form $f$ on a vector space $V$ with basis $\{ e_{1},..., e_{n} \}$ can be represented by an $n \times n$ matrix $F=[f_{ij}]$ where $f_{ij}=f(e_{i},e_{j})$. Then if $v, w \in V$, $f(v,w) = w^{T}Fv$, where $T$ is the transpose.
Is there an analogous matrix representation for a multilinear $k$-form $T: V \times.... \times V \mapsto \mathbb{R}$ ?
Not a matrix, but you can represent it by the $n^k$ numbers $$T_{i_1\cdots i_k}\doteq T(e_{i_1},\ldots,e_{i_k}), $$ where $1\leq i_1,\ldots, i_k\leq n$. If $(e^1,\ldots,e^n)$ is the dual basis, we have $$T =\sum_{i_1,\ldots,i_k=1}^n T_{i_1\cdots i_k} e^{i_1}\otimes\cdots \otimes e^{i_k}, $$ where $\otimes$ denotes tensor product. Meaning the map defined by $$(e^{i_1}\otimes \cdots\otimes e^{i_k})(v_1,\ldots,v_k) = e^{i_1}(v_1)\cdots e^{i_k}(v_k). $$