Linking Number and Green Theorem

Question

I try to solve an equation as $$ Lk(\alpha,\beta)=\int_{\beta}^{}\int_{\alpha}^{} \frac{p-p'}{||p-p'||^{3}}\cdot dp \times dp'=4\pi m$$ and we know that $$ \frac{p-p'}{||p-p'||^{3}}\cdot dp \times dp' = det((p-p'),dp,dp')$$ thus I converted the equation to $$ Lk(\alpha,\beta)=\int_{\beta}^{}\int_{\alpha}^{}\frac{det((p-p'),dp,dp')}{||p-p'||^{3}} $$ I am studying with 3 dimension but I do not know how to write the dp and dp' with 3 components in determinant. Thanks to help.