Your last formula corresponds to the CP (canonical-polyadic) tensor decomposition. This is in a way the equivalent of the SVD for matrices, but its numerical properties aren't as nice: There is no direct way to compute it, and iterative methods often have a tendency to get stuck in local minima.
Another generalization of the SVD to tensors is the HOSVD (higher-order SVD), which consists of orthogonal matrices U, V, and W, and a core tensor K. This tensor is not diagonal like the matrix S in the SVD, but all the slices K(:, :, i) have decreasing Frobenius norm (same for K(:, i, :) and K(i, :, :)). This can be computed directly by applying the SVD to reshaped versions of the input tensor.
