Is there a explicit formula for Levy's area in terms of signature for higher dimension?

Question

It is konwn that levy area for two dimenaional path in terms of signature is given by $A=.5*(S(x)_{a,b}^{1,2}-S(x)_{a,b}^{2,1})$ as described in equation 1.35 in the reference below. https://arxiv.org/pdf/1603.03788.pdf The question, is there a explicit formular for area in terms of signature for higher dimensions? It should be derivable but not sure how? Any reference or help would be highly appreciated.