I am having troubles proving that for a symmetric non degenerate bilinear form $\varphi$ on a finite dimensional vector space $V$, with $\dim V>1,$ there is a basis of $V$ $e_1,...,e_n$ such that the matrix $B$ associated to $\varphi$ with respect to $e_1,...,e_n$ is the anti-diagonal matrix. So far I've tried this: Assume that
(a) there exist a vector $x \in V$ different from zero such that $\varphi(x,x)=0$
(b) there exist an $y \in V$ such that $\varphi(x,y)=1$ and $\varphi(y,y)=0$
I try to show the fact by induction on the dimension $n$:
base of induction: $\dim V=2$ choose a basis $e_1,e_2$ for $V$ for assumptions (a) and (b) $\varphi(e_1,e_1)=\varphi(e_2,e_2)=0$ and $\varphi(e_1,e_2)=\varphi(e_2,e_1)=1$.
Inductive Hypothesis: suppose holds for $n-1=\dim V $.
Consider $V$ such that $\dim V=n$ choose a basis $e_1,...,e_n$ for $V$ then for all $x,y \in V$ we have:
$\varphi(x,y)=\varphi(\sum_{i=1}^n x_ie_i,\sum_{j=1}^n y_je_j)=\sum_{i,j=1}^n x_iy_j \varphi(e_i,e_j)= \sum_{i=1}^n x_iy_n\varphi(e_i,e_n) + \sum_{j=1}^n x_n y_j \varphi(e_n,e_j)$ + $\sum_{i,j=1}^{n-1} x_iy_j \varphi(e_i,e_j)$.
By the inductive hypothesis the latter term has a diagonal form, what can I say for the remaining two terms? How to prove that the remaining terms are all zeroes besides the first for the first term? (for the second term should already follow by symmetry)
We must assume that the underlying field $k$ is algebraically closed for the existence of your $x$ and $y$ as you defined above.
Define $f_1:V\rightarrow k$, $f_2:V \rightarrow k$ by $f_1(z)=\varphi(x,z)$ and $f_2(z)=\varphi(y,z)$. Define $V'=\ker(f_1)\cap\ker(f_2)$.
Define $F(z)=(f_1(z),f_2(z))$. Notice that $F$ has rank 2, since $F(x)=(0,1), F(y)=(1,0)$, and $\ker(F)=V'$. Thus, $\dim(V')=\dim(V)-2$.
Use induction on the dimension of the space in order to find a basis for $V'$ such that $\varphi$ is anti-diagonal w.r.t. this basis. Let $\{e_2,\ldots,e_{\dim(V)-1}\}$ be this basis. Let $e_1=x$ and $e_{\dim(V)}=y$.
Finally, notice that $\varphi$ w.r.t. $\{e_1,\ldots,e_{\dim(V)}\}$ is anti-diagonal.