We prove an analogue of the Sylvester theorem for the generator matrices of the quantum affine algebra ${ \rm U}_q(\hat{ \mathfrak{ gl}}_n)$. We then use it to give an explicit realization of the skew representations of the quantum affine algebra. This allows one to identify them in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand-Tsetlin character (or $q$-character). We also apply the quantum Sylvester theorem to construct a $q$-analogue of the Olshanski algebra as a projective limit of certain centralizers in ${ \rm U}_q(\mathfrak{ gl}_n)$ and show that this limit algebra contains the $q$-Yangian as a subalgebra.
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