For a point in the projective space $\mathbb{P}^n$, $p = [a_0:a_1:\dots :a_n]$, how to see that $$I(p) = \left < x_ia_j - x_ja_i: 0 \leq i \leq n, 0 \leq i \leq n \right> \subseteq K[x_0, \dots,x_n].$$
How does having points from the projective space make this any diffferent from a regular problem? Isn't having the generators from the $K[x_0, \dots, x_n]$ enough?
The point is that the coefficients $a_i$ are defined only up to scalar, so you have to check that the definition makes sense. But this is okay since if you consider an equivalent tuple (i.e. obtained by multiplying all the $a_i$ by a non-zero scalar $ \lambda \in K$) then the new generators for $I(p)$ are just multiples of the old ones by $\lambda$, which is invertible in $K$. Then they define a same ideal.