This is a problem from the book "Finite Dimensional Vector Spaces" by Paul R. Halmos:
If $w$ a $k$-linear form, and if the characteristic of the underlying field of scalars is different from 2 (that is, if $1+1 \ne 0$), then $w$ is the sum of a symmetric $k$-linear form and a skew-symmetric one.
I know how to prove this for a bilinear form $w$. Let $a = \dfrac{w + w^T}{2}$ and $b = \dfrac{w - w^T}{2}$, then $a$ is symmetric and b is skew-symmetric. How can I prove this for any $k$-linear form?
$k$-linear form on $V$ is a multilinear form on $V_1 \bigoplus \cdots \bigoplus V_k$ where $V_1 = \cdots = V_k = V$
I'm thinking about making a symmetric form $w_{sym} = \sum_{\pi \in S_k}{\pi w}$ and skew-symmetric $w_{skew} = \sum_{\pi \in S_k}{(\mathbb{sign} \pi) \pi w}$ ...