Why does there exist a continuous map with no fixed point $f\colon S^n\to S^n$ when $n\ge 1$?

Question

Why does there exist a continuous map with no fixed point $f\colon S^n\to S^n$ when $n\ge 1$?

I can find a continuous map that has no fixed points for the case $n=1$ but I fail to see how this generalizes.

Answer 1

Take $f$ to be the antipodal map.