Group Key Exchange (GKE) is an important tool to develop secure multi-user applications such as group text messages, ad-hoc networks, and so on. Most of the currently deployed GKE schemes are synchronous, i.e., they require all the participants to be online during their execution. However, with more battery-powered devices being used in such applications, the synchronicity requirement is challenging to fulfill. To fill the gaps, asynchronous GKE schemes have been introduced in the literature. Nevertheless, the currently available asynchronous and synchronous GKE schemes rely on Trusted Third Parties (TTPs) for key establishment and management. To this end, reliance on TTPs is a serious shortcoming since TTPs are well known to be the single point of failure. Furthermore, the existing GKE schemes require participants to perform all computations, which can degrade the performance of resource-constrained devices such as Internet of Things (IoT) devices. To solve these problems, in this paper, we propose an asynchronous GKE scheme that uses blockchain and smart contracts to store the security keys-related material and reduce the computational load of the participants. Furthermore, our proposed scheme provides Perfect Forward Secrecy (PFS) and Post-Compromised Security (PCS). Our implementation on Ethereum shows that the proposed scheme can scale to more than 100 participants when combined with a distributed storage system.