I'm working on Grassmanians, specifically on the Plucker embeding $p:G_{d,V}\rightarrow P(\wedge^2(V))$ where $V$ is a $k$-vector space 3-dimensional. Say {v_1,v_2,v_3} is a basis for $V$. Then we can see $P(\wedge^2(V))$ as the projective space $P^2$.
Consider a basis for $\wedge^2(V)$ given by $\{v_1\wedge v_2,v_2\wedge v_3, v_1\wedge v_3\}$. Now I must index a basis for $P^2$ following the basis for $\wedge^2(v)$. There is where I'm having problems. The projective space $P^2$ has dimension 2, so what would it be its indexed projective basis?