For each $n\geq 2$, we show that the class of all finite $n$-dimensional partial orders, when expanded with $n$ linear orders which realize the partial order, forms a Fraïssé class and identify its Fraïssé limit $(D_n,<,<_1,\ldots,<_n)$. We give a finite axiomatization of this limit which specifies it uniquely up to isomorphism among countable structures. We then show that the aforementioned class of finite structures satisfies the Ramsey property and conclude, by the Kechris-Pestov-Todorčević correspondence, that the automorphism group of its Fraïssé limit is extremely amenable. Finally, we identify the universal minimal flow of the automorphism group of the reduct $(D_n,<)$.