How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

Question

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. But I run into the problem that I am actually not sure how the correspondence between the variable in my polynomials (the algebra side) and my projective space (the geometry) is made. This is how I see things now:

When given a vector space $V$ over a field $K$, we choice a basis $\{ v_1, \ldots, v_n\}$. This gives us a set of maps $$x_i:V\to K: \sum_{i=1}^na_iv_i\mapsto a_i$$ Now one can associate to a polynomial $p\in K[x_1,\ldots,x_n]$ a map $\tilde{p}:V\to K$ by setting $$\tilde{p}(v)=\sum_{I}c_ix_{I_1}(v)\cdots x_{I_n}(v)$$ For a projectivized space one now restricts as usual to homogeneous polynomials and then one can talk about their zero's.

And hence the correspondence between our geometry and algebra ultimately depends on a choice of basis for our space. But this raises two questions:

1. Is the zariski topology, which is defined in terms of solutions to systems of polynomials, canonical? i.e. do we get the same topology if we would have chosen a different basis and expressed our polynomials w.r.t. the coordinates induced by this basis.
2. Is there any way of describing a polynomial in a coordinate independent way? So if I have an object sitting in $\mathbb{P}(\bigwedge^k V)$ (of course I'm talking about the Plucker embedding of the Grassmannian), can I describe this object as a variety without choosing coordinates on $\bigwedge^k V$?

The answer to my first question is clearly 'yes, the zariski topology is canonical, since the transition from one set of coordinates to another is given by polynomial equations'. I would still like someone's input on this question though, since I've never seen it discussed anywhere; no one ever shows this or talks about this, which makes me doubt that this is a question that makes sense.

The answer to my second question I think is 'no, there is no way to do this without choosing coordinates', but again I would like someone's input on this to maybe get some more insight.