We define a higher-dimensional analogue of symplectic Khovanov homology. Consider the standard Lefschetz fibration of a 2n-dimensional Milnor fiber of the A_{k-1} singularity. We represent a link by a k-strand braid, which is expressed as an element of the symplectic mapping class group. We then apply the higher-dimensional Heegaard Floer homology machinery to this element. We prove its invariance under arc slides and Markov stabilizations, which shows that it is a link invariant.