The paper presents an approach for effective skin lesion segmentation from dermatoscopic images. Aiming at transferring the weights trained from a network originally designed for image classification task, this study proposes to utilize the first layers of EfficientNet as the encoding layers of a U-Net based architecture. Besides, we introduce an encoder-decoder double skip connection scheme, a new skip connection architecture for extracting useful spatial details of skin lesions from the encoding layers. By the double skip scheme, the approach not only fuses information from the encounter layer in the encoder path to the corresponding layer in the decoder path, but also takes into account information of the proceeding encoding layer. In addition, we propose a new decoder network using the Residual blocks and Convolutional Block Attention Module (CBAM) blocks to handle the gradient vanishing problem as well as penalize the weight of each layer. The proposed Encoder-Decoder Double Skip with the Unet architecture, namely EDDS-Unet, has shown promising performance when evaluated on the official ISIC 2017 challenge and the PH2 databases. The proposed method achieves high evaluation scores with the Dice Similarity Coefficients of 0.907 for the ISIC 2017 and 0.950 for the PH2 databases without pre-or post-processing steps.