Atlas on the Grassmannian Variety

Question

Let $G(k,n)$ the set of all $k$-dimensional sub-spaces of a vector complex space $V$ of dimension $n$.
I know that it is possible to define the grassmannian as the quotient of $\chi(n,k)$ by $GL(k)$ where $\chi(k,n)$ is the set of all $n \times k$ complex matrices with maximal rank with the complex topology and $GL(k)$ is the set of all invertible complex matrices of dimension $k$.
In this way we can give the quotien topology to $G(k,n)$. Than we can build an atlas for the grassmannian that is $$U_I=\{ A\in \chi(k,n):det(A_I)\neq0\}$$ where the set $I=(i_1,...,i_k)$ with $i_r=1,...,n$ such that $i_r \neq i_s$ if $r \neq s$ and $A_I$ the sub-matrix of $A$ with rows selected by $I$. The images of this sets $U_I$ in the quotien are the domains of our chart. For the charts of the atals see page 2 of this pdf fileenter link description here.
I know that via the Plucker's application we can see the set $G(k,n)$ as a sub set of some projective space of arbitrary dimension.
HERE MY QUESTION: clearly due to the fact the plucker aplication is an embedding, the image of the grassmannian via this application is a regolar sub-variety of the projective space. So is it true that the atlas on the embedded grassmannian variety is just the restriction of the standard projective atlas to my sub-variety? If yes how can i write the charts?