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Let $omega=sum_{i=1}^n dx_iwedge dy_iinbigwedge^2mathbb{R}^{2n}$ be a standard symplectic form. The following result is due to Lefschetz:

For $kleq n$, the Lefschetz operator
$L^{n-k}:bigwedge^kmathbb{R}^{2n}to
bigwedge^{2n-k}mathbb{R}^{2n}$ defined by $$
L^{n-k}alpha=alphawedgeunderbrace{omegawedgeldotswedgeomega}_{n-k}=alphawedgeomega^{n-k}
$$ is an isomorphism.

This follows from Proposition 1.2.30 in Huybrechts' Complex Geometry. The proof is slick and it uses representation theory.

Question: What are other standard references for this result? I want to quote it properly and I would like to see a reference to a
proof by brute force. A proof by brute force is not difficult but ugly
and perhaps in some reference I can find an elegant presentation of
such a proof.

This result seems so fundamental that it should be available in many textbooks, but I am not aware of any except the one by Huybrechts.

There is an elementary proof in our 2003 book Exterior Differential Systems and Euler-Lagrange Partial Differential Equations (Bryant, et al, University of Chicago Press). It does not use any representation theory and is not 'brute force'; it only takes a couple of paragraphs using elementary facts about exterior algebra. See Proposition 1.1a, with the proof on page 13.

I learned that proof from Eugenio Calabi more than 30 years ago, and he told me that he had found it sometime back in the 50s.