Assume that $n>1$.

The configuration space of $S^n$ is defined as follows $$M_n=\{(x,y)\in S^n\times S^n\mid x \neq y\}$$

We have two questions:

1.Is there a continuous function $f:M_n \to S^{n-1}$ with $f(y,x)=-f(x,y)$?

2.Is there a continuos function $h;S^n \times S^n \to \mathbb{R}^n$ such that $h(x,y)=-h(y,x) $ and $h(x,-x)\neq 0$

If the answer to either of these two questions is "yes", we can provide an alternative proof for the Borsuk Ulam theorem, inductively. Because an equivalent formulation of the Borsuk Ulam theorem is that:

There is no an odd continuous function $g:S^{n+1}\to S^n$

Assuming that the answer to either of the above two questions is affirmative, we give a proof for the equivalent formulation of the Borsuk Ulam theorem as follows:

Assume that $g:S^{n+1}\to S^n$ is an odd continuous function. then $f(g(x),g(-x))$ ( or $h(g(x),g(-x))$ ) is an odd continuous function from $S^{n+1}$ to $S^{n-1}$ ( or to $\mathbb{R}^n \setminus \{0\}$). This obviously gives a contradiction by induction.