Can we generate a block skew-symmetric matrix by a operator

Question

Suppose we have following speical block matrix $$ X = \begin{bmatrix} 0 & X_3 & -X_2\\ -X_3& 0& X_1\\ X_2 &-X_1 &0 \end{bmatrix} $$ where $X1,X2,X3 \in R^{n \times n}$ and not invertible.Can we generate $X$ by a oprator $\mathcal{A}$ of $[X_1,X_2,X_3]$ or $\begin{bmatrix}X_1 \\ X_2 \\X_3\end{bmatrix}$,so we don't need to store $X$,only need to store $X_1,X_2,X_3$ and obtain $X$ through $X_1,X_2,X_3$.