Let $k$ be a field of characteristic $0$. Consider the $\mathbb N$-graded (polynomial) ring $R=k[x_1,...,x_n]$ graded in the standard way that is the $d$-th graded piece $R_d$ is the $k$-vector space spanned by $\{x_1^{a_1} ...x_n^{a_n} : a_1+...a_n=d \}$ (with $R_0=k$ ) . Now consider the homogeneous ideal $I=(x_1^2,...,x_n^2)$ of $R$ then $R/I$ has a natural $\mathbb N$-grading and for $d\le n$, we have $\dim_k (R/I)_d = {n \choose d} $ and $(R/I)_d=0,\forall d>n $ . (Notice that the Hilbert series of $R/I$ is $(1+t)^n$ )
Now my question is: For $d \le n$, Can we naturally identify the graded pieces (as $k$-vector space) $(R/I)_d$ in terms of the ideals $I$ and $\mathfrak m=(x_1,...,x_n)$ ? Like can we write down something like $(R/I)_d \cong (\mathfrak m^d +I)/(\mathfrak m^{d+1} +I) $ or $(R/I)_d \cong (\mathfrak m^d + I)/\mathfrak m (\mathfrak m^{d} +I) $ , where the isomorphisms are as $k$ vector spaces ?
My motivation comes from the fact that for the polynomial ring $R$ we can actually write $R_d \cong \mathfrak m^d/\mathfrak m^{d+1} $ as $k$-vector spaces.