Let $V$ and $W$ be two real vector spaces. I would like to show that two symmetric and multi-linear functions $$\alpha_1,\alpha_2:V^n\to W$$ are equal if and only if $$\forall v\in V:\alpha_1(v)=\alpha_2(v),$$ where $\alpha(v):=\alpha(v,\ldots,v)$.
The case $n=2$ is easy as $\alpha_1(v,w)=\alpha_2(v,w)$ follows from $\alpha_1(v+w,v+w)=\alpha_2(v+w,v+w)$ (note that scalar multiplication comes into play here). I am not sure how to go to higher dimensions though (induction?). Note that we can not use the polarization formula $$\alpha(v_1,\ldots,v_n)=\frac{1}{n!}\frac{\partial^n}{\partial t_1\cdots\partial t_n}\alpha(t_1v_1+\ldots+t_nv_n)$$ in Nicolaescus's notes as $W$ shall not be finite-dimensional. Is there perhaps a similar formula without derivatives?
Motivation:
I would like to show that the requirement $$\forall(f_0,\ldots,f_n)\in C^\infty(M)^{n+1}:\left\{\prod_{i=0}^n\mathrm{ad}(f_i)\right\}T=0$$ for differential operators of order $n$ is equivalent to $$\forall f\in C^\infty(M):\mathrm{ad}(f)^{n+1}T=0.$$