Here is a positive answer for $n=2.$ The argument doesn't seem to generalize easily to higher $n.$ The idea is to write $df\in X$ as $\star df_1 + df_2\neq 0$ and make use of [Helmholtz decomposition][1].

The set $X$ consists of the matrices not of the form $(\begin{smallmatrix}a&b\\b&-a\end{smallmatrix}),$ in the $(dx,dy)$ basis (for matrices with $\sigma_1=\sigma_2,$ the middle part of an SVD is a positive scalar, so the matrix has to be a scalar multiple of an orthogonal matrix). So we want to find an approximating sequence $g^{(n)}$ with

$$\star dg^{(n)}_1 + dg^{(n)}_2\neq 0$$
on the unit ball, with the usual Hodge star operator $\star(a\;dx + b\;dy)=-b\;dx+a\;dy.$.

We can assume $f$ extends to a function $\bar f$ in $C^\infty(\mathbb R^2,\mathbb R^2)\cap W^{1,2}(\mathbb R^2,\mathbb R^2),$ for example using the Sobolev extension theorem plus a standard result on density of smooth functions in Sobolev spaces.
The combination $\zeta=\star d\bar f_1+d\bar f_2$ is a smooth $L^2$ vector field. By a perturbation we can assume that $\zeta$ is non-zero except at isolated points. By pushing these out of the unit ball - composing with a suitable smooth diffeo $\mathbb R^2\to\mathbb R^2$ that affects only a region of small measure - we can approximate $\zeta$ in $L^2$ by a sequence of smooth $L^2$ vector fields $\gamma_n$ such that $\gamma_n\neq 0$ everywhere in the unit ball.

Each $\gamma_n$ has an orthogonal Helmholtz decomposition which we can write as $\gamma_n=\star dg^{(n)}_1 +dg^{(n)}_2$ where $g^{(n)}_1,g^{(n)}_2$ are smooth functions (possibly not in $L^2$), determined up to additive constants. The functions $g^{(n)}_i$ are unique if we add the requirement $\int_{\mathbb D^2} g^{(n)}_i=\int_{\mathbb D^2} f_i$ for $i=1,2.$ Because Helmholtz decomposition is an orthogonal decomposition, $\star dg^{(n)}_1+dg^{(n)}_2\to \star d\bar f_1+d\bar f_2$ in $L^2$ implies $dg^{(n)}\to d\bar f$ in $L^2.$ The Poincaré–Wirtinger inequality then gives $g^{(n)}|_{\mathbb D^2}\to f$ in $W^{1,2}(\mathbb D^2,\mathbb R^2).$


  [1]: https://en.m.wikipedia.org/wiki/Helmholtz_decomposition