Plücker Relations

Yes, the Plücker relations are written down totally explicitly in terms of the polynomials you require on page 110, equation (3.4.10), of Jacobson's book Finite-Dimensional Algebras over Fields. The proof, attributed by the author to Faulkner (a student of his?), is completely down-to-earth: no identifications, no duality,...

Edit Since Martin doesn't have access to the book, I'm adding an online presentation, with the relevant equations on page 21. It is very elementary, with concrete examples, and might appeal to readers whose interest has been whetted by Martin's question.
And the bibliography contains a reference to a masterful article by Kleiman and Laksov, which also contains the Plücker relations handled with minors of determinants and nothing else.

I think the answer provided above could be improved a bit. (Not logically, just made a bit clearer.) My method avoids having extra variables and setting repeating sequences equal to zero.

Take a field $F$, and let $$X = \{x_H : H \subset \{1,2,...n\} \text{ and } |H| = d\}$$ So we can think of variables as being indexed by length $d$ increasing sequences. For $I,K \subset \{1,2,...,n\}$ of size $d-1$ and $d+1$, respectively, we can define the $(I,K)$ Plucker relation $Pl_{I,K}(X)$ by the formula:$$ Pl_{I,K}(X) = \sum_{k \in K - I} (-1)^{S_{I,K}(k)} x_{I \cup \{k\}} x_{K - \{k\}} $$ And the sign $S_{I,K}(k) = \#\{i \in I : k < i\} + \#\{\ell \in K: k < \ell\} $