TITLE: Non-existence of nowhere vanishing vector field on even dimensional sphere
QUESTION [5 upvotes]: I want to show that for even dimensional spheres, there does not exist a nowhere vanishing vector field, namely a non-trivial cross section of its tangent bundle. I am wondering how elementary the proof could be. While I know this should follow from Lefschetz fixed-point theorem (I have seen the proof when $n=2$ a long time ago), I believe there should be easier proofs available. 
It seems this theorem follows from standard differential topology tools (like Poincare-Hopf theorem and computing the Euler characteristic). And similarly one may use tools like Euler class. But is there a more elementary proof available (at the level of singular homology and cohomology)?
Here is a "cyclic type argument". Assuming I can show that if a nonwhere vanshing vector field exists, then the antipodal map must be homotopically equivalent to the identity map, then I would be able to prove the original statement since the antipodal map for even dimensional sphere has degree $-1$. However, I do not know how to prove the first statement without passing to even and odd cases. So this does not really help very much. 
Sorry if this question is too trivial. I am looking for a hint (not an answer). 
Reference:
Milnor&Stasheff, Problem 2-B
http://en.wikipedia.org/wiki/Hairy_ball_theorem

REPLY [3 votes]: (Posting an answer to get this off the unanswered queue.)
Ralph Mellish mentions that proof of the kind requested can be found as Cor 1.26 in Vick's Homology Theory.
I note the same proof appears as theorem 2.28 in Chapter 2.2 of Hatcher's Algebraic Topology.