How to calculate the Snake Index in N dimensions?

Question

I have written the following code in c# where it calculates the Snake index of a 2-dimensional point.

    public static uint SnakeCurveIndex(uint bits, uint x, uint y )
    {

        uint index = 0;
        //The dimension of the array
        uint dim = (uint)Math.Pow( 2.0 , (double)bits);

        if(y % (uint)2 == 0 )
        {
            index = x + y * dim;

        }
        else
        {
            index = (dim - 1 - x) + y * dim;
        }
        if (index >= dim*dim)
        {
            //Debug console 
            throw new Exception("The index is out of bounds");
        }

        return index;
    }

The variable bits it responsible for the order of the curve. The following image represents the order of the curve for 1 to 3.

My question is who to extend this code for n-dimensional points? Do I need a multidimensional-array or some other technique?

Thank you for your time.